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Three topics in algebraic curves over finite fields

Mariana de Almeida Nery CoutinhoTese de Doutorado do Programa de Pós-Graduação emMatemática (PPG-Mat)

SERVIÇO DE PÓS-GRADUAÇÃO DO ICMC-USP

Data de Depósito:

Assinatura: ______________________

Mariana de Almeida Nery Coutinho


Doctoral dissertation submitted to the Institute ofMathematics and Computer Sciences – ICMC-USP, inpartial fulfillment of the requirements for the degree ofthe Doctorate Program in Mathematics. FINALVERSION

Concentration Area: Mathematics

Advisor: Prof. Dr. Herivelto Martins Borges Filho

USP – São CarlosMay 2019

Ficha catalográfica elaborada pela Biblioteca Prof. Achille Bassi e Seção Técnica de Informática, ICMC/USP,

com os dados inseridos pelo(a) autor(a)

Bibliotecários responsáveis pela estrutura de catalogação da publicação de acordo com a AACR2: Gláucia Maria Saia Cristianini - CRB - 8/4938 Juliana de Souza Moraes - CRB - 8/6176

C871tCoutinho, Mariana de Almeida Nery Three topics in algebraic curves over finitefields / Mariana de Almeida Nery Coutinho;orientador Herivelto Martins Borges Filho. -- SãoCarlos, 2019. 150 p.

Tese (Doutorado - Programa de Pós-Graduação emMatemática) -- Instituto de Ciências Matemáticas ede Computação, Universidade de São Paulo, 2019.

1. Plane and space curves. 2. Curves over finitefields. 3. Rational points. 4. Zeta functions. 5.Automorphisms. I. Borges Filho, Herivelto Martins,orient. II. Título.

Mariana de Almeida Nery Coutinho

Três tópicos em curvas algébricas sobre corpos finitos

Tese apresentada ao Instituto de CiênciasMatemáticas e de Computação – ICMC-USP,como parte dos requisitos para obtenção do títulode Doutora em Ciências – Matemática. VERSÃOREVISADA

Área de Concentração: Matemática

Orientador: Prof. Dr. Herivelto Martins Borges Filho

USP – São CarlosMaio de 2019

Para Luciana, Marcelo e Marcelinho.

ACKNOWLEDGEMENTS

O verbete gratidão é proveniente da palavra latina grafia, a qual significa graça, ouda palavra gratus, que traduz-se como agradável. Por extensão, gratidão é o reconhecimentoagradável por tudo quanto se recebe ou se lhe é concedido. Assim, nessa parte agradeço (isto é,manifesto a minha gratidão) a todos que contribuíram, direta ou indiretamente, para que essetrabalho pudesse ser escrito.

Ao professor Herivelto, pelo imenso apoio, paciência e atenção ao longo dos últimosquatro anos; pelo exemplo de pessoa, professor e orientador; pelo entusiasmo que semprecolocou em cada explicação e questão a ser analisada; mas, especialmente, por ter me mostradoa importância de se fazer perguntas e de não se ter medo de errar.

A cada um dos professores do ICMC com os quais tive a oportunidade conviver. Emespecial, agradeço ao professor Sérgio Monari, pelo acolhimento em todos esses anos, e aoprofessor Daniel Levcovitz, por ter me aceitado como aluna tantas vezes e pela enorme ajuda.

Aos professores Abramo Hefez, Daniel Levcovitz (novamente) e Fernando Torres, pelaparticipação na banca que examinou esse trabalho, bem como por cada uma das sugestões ecomentários que contribuíram para a melhoria desse texto.

À equipe da cantina do ICMC, pela alegria com que sempre me recebeu.

Aos amigos da minha turma de doutorado, Alex (Francisco e Silva), Angelina, Camila,Carol, Jean, Liliam e Pedro, que de diversas formas muito especiais marcaram essa trajetória.

Aos amigos “INCA” (Interessados Nas Curvas Algébricas), Alex, Cirilo, Grégory, Lucas,Nazar, Pietro e Roberto, por todo apoio e tudo que me ensinaram.

Aos amigos do ICMC que não se enquadram nos dois grupos acima, Cesar, Eduardo,Hellen, Jackson, Marielle, Pryscilla, Renan e Wilker.

Aos professores do Departamento de Matemática da UFJF. Em especial agradeço àprofessora e orientadora Beatriz Motta pelo imenso incentivo para que eu pudesse iniciar odoutorado, bem como pela presença e apoio nesses últimos quatro anos.

Aos professores do Departamento de Física da UFJF. Em especial agradeço ao professorMaikel Ballester pela enorme contribuição para a minha formação acadêmica e pessoal enquantosua aluna de iniciação científica.

Aos amigos da UFJF, Adriele, Janaina, Leandro, Sandra e Santiago, que mesmo com adistância sempre estiveram ao meu lado.

Ao amigo Eli Vilela, pelas conversas sobre Matemática, violão, piano e diversos outrosassuntos.

À minha família (na mais ampla acepção desse termo), simplesmente por tudo.

Aos amigos da FEAK, pelo imenso carinho.

Às minhas amigas de infância, por todos os momentos especiais.

Ao amigo Bruno Marques, pelo apoio incondicional.

A cada um dos motoristas de ônibus que com o seu trabalho permitiram que a distânciaentre Juiz de Fora e São Carlos ficasse um pouco menor.

À Coordenação de Aperfeiçoamento de Pessoal de Nível Superior – (Brasil) CAPES –Código de Financiamento 001, e ao CNPq – Processo 154359/2016-5, pela bolsa de doutorado.

“A ciência é neta da curiosidade e filha do estudo.”

(Emmanuel)

ABSTRACT

COUTINHO, M. A. N. Three topics in algebraic curves over finite fields. 2019. 150 p.Tese (Doutorado em Ciências – Matemática) – Instituto de Ciências Matemáticas e de Computa-ção, Universidade de São Paulo, São Carlos – SP, 2019.

Inserted in the context of algebraic curves defined over finite fields, the present thesis addressesthe study of the following three topics: plane sections of Fermat surfaces over finite fields;bounds for the number of Fq-rational points on aXdY d−Xd−Y d + b = 0 and the number ofchords of an affinely regular polygon inscribed in a hyperbola passing through a given point; thenumber of Fqn-rational points, the L-polynomial and the automorphism group of the generalizedSuzuki curve.

Keywords: Plane and space curves, Curves over finite fields, Rational points, Zeta functions,Automorphisms.

RESUMO

COUTINHO, M. A. N. Três tópicos em curvas algébricas sobre corpos finitos. 2019. 150p. Tese (Doutorado em Ciências – Matemática) – Instituto de Ciências Matemáticas e deComputação, Universidade de São Paulo, São Carlos – SP, 2019.

Inserida no contexto das curvas algébricas definidas sobre corpos finitos, a presente tese abordao estudo dos seguintes três tópicos: seções planas das superfícies de Fermat sobre corposfinitos; cotas para o número de pontos Fq-racionais em aXdY d−Xd−Y d +b = 0 e o número decordas passando por um determinado ponto de um polígono afinamente regular inscrito em umahipérbole; o número de pontos Fqn-racionais, o L-polinômio e o grupo automorfismos da curvade Suzuki generalizada.

Palavras-chave: Curvas planas e espaciais, Curvas sobre corpos finitos, Pontos racionais,Funções zeta, Automorfismos.

LIST OF FIGURES

Figure 1 – A comparison with the Hasse-Weil bound . . . . . . . . . . . . . . . . . . 119

LIST OF TABLES

Table 1 – The number of Fq-rational points on C with i ∈ {1,2} zero coordinates. . . . . . . 90Table 2 – The number N(1)+N(2)+N(3). . . . . . . . . . . . . . . . . . . . . . . . . . . 92Table 3 – Linear components of C for e0e1e2 = 0. . . . . . . . . . . . . . . . . . . . . . . 95Table 4 – Curve F for d odd. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Table 5 – Curve F for d even. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101Table 6 – Fq-points with zero coordinates on C for d odd . . . . . . . . . . . . . . . . 103Table 7 – Fq-points with zero coordinates on C for d even . . . . . . . . . . . . . . . 103

CONTENTS

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

I FOUNDATIONS 25

1 PROJECTIVE ALGEBRAIC CURVES . . . . . . . . . . . . . . . . . 271.1 Plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.1.1 First definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 271.1.2 Branches of plane curves . . . . . . . . . . . . . . . . . . . . . . . . . . 301.1.2.1 Branch representations and branches . . . . . . . . . . . . . . . . . . . . . 301.1.2.2 Order and tangent of branch representations and branches . . . . . . . . . 321.1.2.3 Intersection multiplicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.1.3 Function fields of irreducible plane curves . . . . . . . . . . . . . . . . 341.2 Algebraic function fields . . . . . . . . . . . . . . . . . . . . . . . . . . 351.2.1 Plane models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.2.2 Places . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361.2.3 Some remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371.2.4 Expansion of a function at a local parameter . . . . . . . . . . . . . . 381.2.5 Separating variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391.2.6 Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 391.2.7 Hasse derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411.2.8 Differentials and the notion of genus . . . . . . . . . . . . . . . . . . 421.2.9 Divisors and linear series . . . . . . . . . . . . . . . . . . . . . . . . . . 441.2.10 Linear systems of plane curves . . . . . . . . . . . . . . . . . . . . . . 461.3 Space curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 471.3.1 Rational transformations and morphisms . . . . . . . . . . . . . . . . 501.3.2 Some remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521.3.3 Morphisms × linear series . . . . . . . . . . . . . . . . . . . . . . . . . 531.3.3.1 Morphisms from linear series . . . . . . . . . . . . . . . . . . . . . . . . . 531.3.3.2 Linear series from morphisms . . . . . . . . . . . . . . . . . . . . . . . . . 55

2 PROJECTIVE ALGEBRAIC CURVES DEFINED OVER FINITEFIELDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.1 Plane curves defined over Fq . . . . . . . . . . . . . . . . . . . . . . . 57

2.2 Fq-rational branches of plane curves . . . . . . . . . . . . . . . . . . . 582.3 Fq-rational function fields and Fq-rational places . . . . . . . . . . . . 592.3.1 Some remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602.4 Fq-rational divisors and linear series . . . . . . . . . . . . . . . . . . . 602.5 Space curves defined over Fq . . . . . . . . . . . . . . . . . . . . . . . 612.5.1 Rational transformations and morphisms defined over Fq . . . . . . 622.5.2 Nonsingular models defined over Fq . . . . . . . . . . . . . . . . . . . 622.6 The Zeta function of a curve defined over Fq . . . . . . . . . . . . . 622.7 The Hasse-Weil theorem and maximal curves . . . . . . . . . . . . . 65

3 THE STÖHR-VOLOCH THEORY . . . . . . . . . . . . . . . . . . . 673.1 Morphisms and Weierstrass points . . . . . . . . . . . . . . . . . . . . 673.1.1 Hermitian invariants and osculating spaces . . . . . . . . . . . . . . . 683.1.2 Order sequence and ramification divisor . . . . . . . . . . . . . . . . . 693.2 The Stöhr-Voloch theorem . . . . . . . . . . . . . . . . . . . . . . . . 73

II THREE TOPICS IN ALGEBRAIC CURVES OVER FI-NITE FIELDS 79

4 PLANE SECTIONS OF FERMAT SURFACES OVER FINITE FIELDS 814.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.3 Rudiments of the Stöhr-Voloch theory . . . . . . . . . . . . . . . . . 834.4 Points and linear components of curve C . . . . . . . . . . . . . . . . 894.4.1 Points with zero coordinates . . . . . . . . . . . . . . . . . . . . . . . 894.4.2 Points without zero coordinates . . . . . . . . . . . . . . . . . . . . . 904.4.3 Linear components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 924.5 Preliminary result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.6 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 984.6.1 Frobenius classicality and absolute irreducibility . . . . . . . . . . . . 984.6.2 Proof of Theorem 4.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 100

5 ON SOME GENERALIZED FERMAT CURVES AND CHORDS OFAN AFFINELY REGULAR POLYGON INSCRIBED IN A HYPER-BOLA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1065.3 The curve F . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1075.3.1 The case q = p . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.3.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5.3.1.2 The proof of Theorem 5.3.4 . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.4 Affinely regular polygons . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.4.1 A brief introduction to affinely regular polygons . . . . . . . . . . . . 1165.4.2 Number of chords of an affinely regular polygon inscribed in a hy-

perbola passing through a given point . . . . . . . . . . . . . . . . . . 118

6 ON THE ZETA FUNCTION AND THE AUTOMORPHISM GROUPOF THE GENERALIZED SUZUKI CURVE . . . . . . . . . . . . . . 121

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.3 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.3.1 L-polynomials and supersingular curves . . . . . . . . . . . . . . . . . 1256.3.2 Elementary abelian p-extensions of algebraic fuction fields . . . . . 1266.3.3 On the number of Fqn-rational points of Y p−Y = XR(X) . . . . . . . 1276.3.4 The curve GS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.3.5 Automorphism group . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1306.4 Proof of Theorem 6.1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.5 Proof of Theorem 6.1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.5.1 Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.6.1 The L-polynomial of XGS

for p = 3 . . . . . . . . . . . . . . . . . . . 1396.6.2 The L-polynomial of XGS






for p = 19 . . . . . . . . . . . . . . . . . . . 1426.7 Proof of Theorem 6.1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . 1426.8 Final remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145

23

INTRODUCTION

The theory of algebraic curves, or the theory of algebraic function fields in one variable,which is the field-theoretic counterpart of algebraic curves, is the result of the efforts of severalmathematicians over the past centuries. A more precise formulation (over the complex field)dates back to the nineteenth century, with the works of R. Dedekind, H. M. Weber, K. Hensel andG. Landsberg, and from a geometric point of view, with the works of Max Noether, A. Clebschand P. Gordan (see (CHEVALLEY, 1951, Introduction)).

In the first half of the twentieth century, further development on the theory of algebraiccurves was provided by the works of E. Artin, H. Hasse, F. K. Schmidt and A. Weil, wherefields other than the complex were also taken into account. This fact led, among other aspects,to the emergence of the theory of algebraic curves defined over finite fields, which appearsin the intersection of important areas of mathematics, such as finite geometry, coding theory,cryptography and number theory, being nowadays a broad issue of study and research.

Divided into two parts, each one containing three chapters, this thesis is inserted in thecontext of algebraic curves defined over finite fields.

Part I establishes the theoretical basis for the study of algebraic curves, as well as thefundamental tools and terminology used in the second part. Presenting the topics as commonlyfound in textbooks, however without providing the proofs, this part was drawn up considering itsuse for a consultation on the subjects and the connections between them, and was not thereforeelaborated aiming at a linear reading of the text. Chapters 1 and 2 are based essentially on(HIRSCHFELD; KORCHMÁROS; TORRES, 2008). Chapter 1 describes the theory of projectivealgebraic curves in arbitrary characteristic. There, a comparison between the concept of placeas constructed in (SEIDENBERG, 1968) and (HIRSCHFELD; KORCHMÁROS; TORRES,2008) and that which is usually considered in the literature (see (GOLDSCHMIDT, 2003) and(STICHTENOTH, 2009) for instance) is made. Chapter 2 gives an introduction to the projectivealgebraic curves defined over finite fields, ending with the definition of the Zeta function andwith the remarkable Hasse-Weil theorem. Finally, Chapter 3 collects the main elements of theStöhr-Voloch theory, which is the basis of two of the three chapters in the second part. It is worthpointing out that, although the issue automorphism group of projective curves is important forthe development of Chapter 6, this topic is not presented in this first part of the thesis, being onlyaddressed in the referred chapter.

Part II is the heart of this work. In it, three independent topics, although related intheir essence to the study of the number of rational points on curves defined over finite fields,

24 Introduction

are considered. The order chosen to present the subjects follows the chronological order ofdevelopment of them.

In Chapter 4, all curves defined over Fq arising from a plane section

P : X3− e0X0− e1X1− e2X2 = 0

of the Fermat surfaceS : Xd

0 +Xd1 +Xd

2 +Xd3 = 0,

where q = pm = 2d + 1 is a prime power, p > 3, and e0,e1,e2 ∈ Fq, are characterized. Inparticular, it is proved that any nonlinear component F ⊆ P ∩S is a nonsingular classical curveof degree d6 d attaining the Stöhr-Voloch bound

#F (Fq)612

d(d+q−1)− 12

i(d−2),

with i ∈ {0,1,2,3,d,3d}.

Now, let F be the projective plane curve defined over Fq, with affine equation given by

aXdY d−Xd−Y d +b = 0,

where q = pm is a prime power and ab /∈ {0,1}. Considering for each s ∈ {2, . . . ,d− 1} thebase-point-free linear series DP1,P2

s cut out on F by the linear system of all curves of degree s

passing through the singular points P1 = (1 : 0 : 0) and P2 = (0 : 1 : 0) of F , Chapter 5 determinesan upper bound for the number Nq(X ) of Fq-rational points on the nonsingular model X of F

defined over Fq in cases where DP1,P2s is Fq-Frobenius classical. As a consequence, when Fq is

the prime field Fp, the bound obtained for Nq(X ) improves in several cases the known boundsfor the number of chords of an affinely regular polygon inscribed in a hyperbola passing througha given point distinct from its vertices.

Last (but not least), for p an odd prime number, q0 = pt and q = pm = p2t−1, Chapter 6studies the nonsingular model defined over Fq of

Y q−Y = Xq0(Xq−X)

from the point of view of its number of Fqn-rational points and automorphism group. As aconsequence, a description of the L-polynomial of this curve is provided.

Part I

Foundations

27

CHAPTER

1PROJECTIVE ALGEBRAIC CURVES

The purpose of this chapter is to present some background on the subject projective alge-

braic curves. Based on (FULTON, 2008, Chapters 3, 5, 7 and Appendix A), (GOLDSCHMIDT,2003, Chapters 1 and 2), (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Chapters 1, 2,4, 5, 6 and 7), (SEIDENBERG, 1968, Chapters 12, 13, 14, 16, 19 and 20), (STICHTENOTH,2009, Chapters 1, 3 and 4) and (TORRES, 2000, Sections 1 and 2), the main concepts to developthe second part of this work are here collected. The way chosen to organize the topics does notfollow in several circumstances the natural order required to prove the results, which are onlystated. Also, in many cases, the objects introduced in the text are assumed to be well definedwithout proofs or other comments in this direction.

Throughout this chapter, K denotes a fixed algebraically closed field of characteristicp> 0.

1.1 Plane curves

1.1.1 First definitions

Definition 1.1.1. We define ∼ as the following equivalence relation on the set of nonzerohomogeneous polynomials in K[X ,Y,Z]:

F(X ,Y,Z)∼ G(X ,Y,Z)⇔ There exists λ ∈K* such that F(X ,Y,Z) = λG(X ,Y,Z).

Definition 1.1.2. Let F(X ,Y,Z) ∈K[X ,Y,Z] be a nonzero homogeneous polynomial of degreed > 1.

1. The projective plane algebraic curve in P2(K)

F : F(X ,Y,Z) = 0,

sometimes denoted only by F , is the equivalence class of F(X ,Y,Z) with respect to ∼.

28 Chapter 1. Projective algebraic curves

2. The degree of F is the positive integer d.

3. If F(X ,Y,Z) = ∏ki=1 Gi(X ,Y,Z)li , with Gi(X ,Y,Z) irreducible over K, then each Gi :

Gi(X ,Y,Z) = 0 is called a component of F and li is its multiplicity. Also, F is irreducible

if F(X ,Y,Z) is an irreducible polynomial over K. Otherwise, F is called reducible.

4. The point set of F , here also denoted by F , is the set{(α : β : γ) ∈ P2(K) : F(α,β ,γ) = 0

}.

Notation 1.1.3. Hereafter, the expression plane curve means a projective plane algebraic curvein P2(K).

Remark 1.1.4. In some cases, a plane curve F is denoted in affine coordinates by

F : F(X ,Y ) = 0,

where F(X ,Y ) ∈K[X ,Y ] is a polynomial of degree d > 1. In this case,

F : F*(X ,Y,Z) = 0

and the point set of F is described as follows:

F =

{(α : β : γ) ∈ P2(K) : F*(α,β ,γ) = 0

}=

{(α : β : 1) ∈ P2(K) : F(α,β ) = 0

}∪{(α : β : 0) ∈ P2(K) : F*(α,β ,0) = 0

},

where {(α : β : 1) ∈ P2(K) : F(α,β ) = 0

}is the set of affine points of F and{

(α : β : 0) ∈ P2(K) : F*(α,β ,0) = 0}

is the set of points at infinity of F . Here, d is also the degree of F , and F is irreducible if andonly if F(X ,Y ) (or equivalently F*(X ,Y,Z)) is irreducible.

Definition 1.1.5. A plane curve is called a line or a conic if its degree is equal to 1 or 2,respectively.

Definition 1.1.6. Let F : F(X ,Y,Z) = 0 be a plane curve.

1. A singular point of F is a point P ∈F such that

∂F∂X

∣∣∣∣P=

∂F∂Y

∣∣∣∣P=

∂F∂Z

∣∣∣∣P= 0.

Otherwise, P is called a nonsingular point of F .

1.1. Plane curves 29

2. If P is a nonsingular point of F , then the tangent line to F at P is given by

TP :∂F∂X

∣∣∣∣PX +

∂F∂Y

∣∣∣∣PY +

∂F∂Z

∣∣∣∣PZ = 0.

3. If all the points of F are nonsingular, then F is a nonsingular plane curve. Otherwise, F

is called a singular plane curve.

The following two properties, that are well established in the literature, relate the nonsin-gularity of a plane curve and its irreducibility.

Proposition 1.1.7. Let F be a plane curve. If F is nonsingular, then F is also irreducible.

Proposition 1.1.8. Let F be an irreducible plane curve. Then F has a finite number of singularpoints.

Remark 1.1.9. Let F : F(X ,Y ) = 0 be a plane curve of degree d. For each P = (α : β : 1) ∈P2(K), write

F(X +α,Y +β ) = Fm(X ,Y )+Fm+1(X ,Y )+ · · ·+Fd(X ,Y ),

where Fi(X ,Y ) ∈K[X ,Y ] is a homogeneous polynomial of degree i, for i =m,m+1, . . . ,d, andFm(X ,Y ) = 0. Also, for m> 1, Fm(X ,Y ) may be written as

Fm(X ,Y ) = ∏Lk(X ,Y )lk ,

where the Lk(X ,Y ) are distinct homogeneous polynomials of degree 1. From this, it follows thatP lies on F if and only if m> 0. Further, P is a nonsingular point of F if and only if m= 1,and in this case

TP : F1(X−α,Y −β ) = 0.

Definition 1.1.10. Consider the notation as in Remark 1.1.9. Then, mP :=m is the multiplicity

of F at P. Further:

1. For mP > 1, the Tk : Lk(X−α,Y −β ) = 0 are the tangent lines to F at P.

2. For mP > 2, P is called an ordinary singular point of F if F has mP distinct tangent linesat P.

Remark 1.1.11. Suppose that the plane curve F is given by the equation F(X ,Y,Z) = 0 andlet P = (α : β : γ) ∈ P2(K). In order to define the multiplicity of F at P, one may proceedas in Remark 1.1.9 by considering the curve F given in affine coordinates by F(X ,Y,1) =0, F(X ,1,Z) = 0 or F(1,Y,Z) = 0 according to whether γ , β or α are different from zero,respectively.


1.1.2 Branches of plane curves

1.1.2.1 Branch representations and branches

Definition 1.1.12. A branch representation ă is a point of P2(K((T )))∖P2(K).

Definition 1.1.13. The coordinates (α(T ) : β (T ) : γ(T )) of a branch representation ă are special

if ordT (α(T ))> 0, ordT (β (T ))> 0, ordT (γ(T ))> 0, and at least one of these orders is equalto zero.

Remark 1.1.14. Every branch representation ă can be written in special coordinates. Indeed,writing ă = (α(T ) : β (T ) : γ(T )) and defining

e =−min{

ordT (α(T )),ordT (β (T )),ordT (γ(T ))},

it follows that

ă = (T eα(T ) : T e

β (T ) : T eγ(T )),

where ordT (T eα(T )) > 0, ordT (T eβ (T )) > 0, ordT (T eγ(T )) > 0, and at least one of theseorders is equal to zero.

Definition 1.1.15. Let ă be a branch representation and suppose that (α(T ) : β (T ) : γ(T )) arespecial coordinates for ă. Then, (α(0) : β (0) : γ(0)) ∈ P2(K) is the center of ă.

Definition 1.1.16. Let ă be a branch representation and let (α(T ) : β (T ) : γ(T )) be specialcoordinates for ă. If γ(T ) = 1 (resp. α(T ) = 1 or β (T ) = 1), then the coordinates (α(T ), β (T ))

(resp. (β (T ), γ(T )) or (α(T ), γ(T ))) are the special affine coordinates of ă with respect to Z

(resp. X or Y ).

Definition 1.1.17. 1. Two elements (ζ1(T ), η1(T )) and (ζ2(T ), η2(T )) of K[[T ]]2 are equiv-

alent if there exists a K-automorphism σ of K[[T ]] such that

ζ1(T ) = σ(ζ2(T )) and η1(T ) = σ(η2(T )).

2. An element (ζ (T ), η(T )) ∈K[[T ]]2 is imprimitive if there exist (ζ1(T ), η1(T )) ∈K[[T ]]2

and a K-monomorphism σ of K[[T ]] that is not a K-automorphism satisfying

ζ (T ) = σ(ζ1(T )) and η(T ) = σ(η1(T )).

Otherwise, (ζ (T ), η(T )) is called primitive.

Definition 1.1.18. 1. Two branch representations are equivalent if they have special affinecoordinates with respect to the same variable (X , Y or Z) that are equivalent.

2. A branch representation is imprimitive if it has imprimitive special affine coordinates withrespect to X , Y or Z. Otherwise, it is called primitive.


Proposition 1.1.19. Let ă be an imprimitive branch representation with imprimitive special affinecoordinates (α(T ), β (T )) with respect to Z. Then, there exists a primitive branch representationă1 with primitive special affine coordinates (α1(T ), β1(T )) with respect to Z such that

α(T ) = σ(α1(T )) and β (T ) = σ(β1(T ))

for some K-monomorphism σ of K[[T ]] that is not a K-automorphism. An analogous situationholds for an imprimitive branch representation that admits imprimitive special affine coordinateswith respect to X or Y .

Proof. The result follows from (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem4.20). �

Definition 1.1.20. Let ă be an imprimitive branch representation and, without loss of generality,suppose that it has imprimitive special affine coordinates (α(T ), β (T )) with respect to Z. Letă1 be a primitive branch representation with primitive special affine coordinates (α1(T ), β1(T ))

with respect to Z satisfying

α(T ) = σ(α1(T )) and β (T ) = σ(β1(T )),

for some K-monomorphism σ of K[[T ]] that is not a K-automorphism. Then, the ramification

index of ă is the order

ordT (σ(T )).

Proposition 1.1.21. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Corollary 4.25). Equiv-alent imprimitive branch representations have the same ramification index. Further, the primitivebranch representations to which they give rise are also equivalent.

Finally, this subsection ends with the notions of branch and branch of a plane curve.

Definition 1.1.22. A branch b is an equivalence class of primitive branch representations. Thecenter of a branch is the center of any of its primitive representations.

Definition 1.1.23. A branch b is a branch of a plane curve F : F(X ,Y,Z) = 0 if its primitiverepresentations ă = (α(T ) : β (T ) : γ(T )) satisfy

F(α(T ), β (T ), γ(T )) = 0.

Theorem 1.1.24. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 4.30). Thecenter of a branch of a plane curve is a point of the curve.

Conversely, the following occurs.


Theorem 1.1.25. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorems 4.32, 4.45 and4.46). Each point P of a plane curve F is the center of at least one and at most mP branches ofF , where mP is the multiplicity of F at P. Also, the following occurs:

1. If P is a nonsingular point of F , then P is the center of a unique branch of F .

2. If P is an ordinary singular point of F , then P is the center of exactly mP branches of F .

Theorem 1.1.26. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 4.37). Let F

and G be distinct irreducible plane curves. Then F and G do not have branches in common.

1.1.2.2 Order and tangent of branch representations and branches

Let (α(T ) : β (T ) : γ(T )) be special coordinates for a branch representation ă. If α(T ),β (T ) and γ(T ) are linearly independent over K, then{

ordT (αα(T )+ββ (T )+ γγ(T )) : (α : β : γ) ∈ P2(K)

}=

{j0, j1, j2

},

where j0, j1 and j2 are non-negative integers satisfying 0 = j0 < j1 < j2. Further, j1 is given by

min{

ordT (αα(T )+ββ (T )+ γγ(T )) : (α : β : γ) ∈ P2(K) and αα(0)+ββ (0)+ γγ(0) = 0},

and

j2 = ordT (α0α(T )+β0β (T )+ γ0γ(T ))

for a unique point (α0 : β0 : γ0) ∈ P2(K).

Definition 1.1.27. 1. The order of ă is the positive number j1.

2. The order sequence of ă is the sequence ( j0, j1, j2).

3. The tangent line of ă is defined by

Tă : α0X +β0Y + γ0Z = 0.

Remark 1.1.28. When α(T ), β (T ) and γ(T ) are linearly dependent, the same previous defini-tions hold by considering j2 = ∞.

Definition 1.1.29. The order, order sequence and tangent of a branch b are, respectively, theorder, order sequence and tangent of any of its primitive representations.

This subsection ends with the following definition.

Definition 1.1.30. A branch is linear if its order is equal to 1.


1.1.2.3 Intersection multiplicity

Definition 1.1.31. Let G : G(X ,Y,Z) = 0 be a plane curve and let b be a branch centered at thepoint P. If (α(T ) : β (T ) : γ(T )) are special coordinates for a primitive representation ă of b,then the intersection multiplicity of G and b is defined by

I(P,G ∩b) :=

{ordT (G(α(T ), β (T ), γ(T ))), if b is not a branch of G

∞, otherwise.

Definition 1.1.32. Let F be a plane curve and let P be a point of F . If G is another plane curve,then the intersection multiplicity of F and G at P is defined by

I(P,F ∩G ) = ∑b is a branch of F

centered at P

I(P,G ∩b).

Remark 1.1.33. From the considerations at the end of Section 4.4 of (HIRSCHFELD; KORCH-MÁROS; TORRES, 2008), the intersection multiplicity of two plane curves F : F(X ,Y,Z) = 0and G : G(X ,Y,Z) = 0 at a point P, as established in Definition 1.1.32, satisfies the followingpostulates:

I 1) I(P,F ∩G ) is a non-negative integer if F and G have no common component through P.

I 2) I(P,F ∩G ) = ∞ if F and G have a common component through P.

I 3) I(P,F ∩G ) = 0 if and only if P /∈F ∩G .

I 4) I(P,F ∩G ) = 1 if F and G are two distinct lines through P.

I 5) I(P,F ∩G ) = I(P,G ∩F ).

I 6) I(P,F ∩ (G +H F )) = I(P,F ∩G ) for any H : H(X ,Y,Z) = 0, where

G +H F : G(X ,Y,Z)+H(X ,Y,Z)F(X ,Y,Z) = 0.

I 7) I(P,F ∩G H ) = I(P,F ∩G )+ I(P,F ∩H ) for any H : H(X ,Y,Z) = 0, where

G H : G(X ,Y,Z)H(X ,Y,Z) = 0.

Since a number satisfying the previous postulates is unique by Theorem (HIRSCHFELD; KO-RCHMÁROS; TORRES, 2008, Theorem 3.8), Definition 1.1.32 coincides with other definitionsfound in the literature, among which it is possible to mention (HIRSCHFELD; KORCHMÁROS;TORRES, 2008, Theorem 3.9 and Definition 3.12) and (FULTON, 2008, Theorem 3 of Chapter3 and considerations on Page 54).

In particular, if F : F(X ,Y,Z) = 0 is an arbitrary plane curve and L : L(X ,Y,Z) = 0 isa line, then I(P,F ∩L ) can be determined as follows.

Without loss of generality, suppose that P = (α : β : 1) and define

F*(X ,Y ) := F(X ,Y,1)

L*(X ,Y ) := L(X ,Y,1).


Then, writing F*(X +α,Y +β ) = FmP(X ,Y )+FmP+1(X ,Y )+ · · ·+Fd(X ,Y ) as in Remark 1.1.9,where d is the degree of F , it follows that

I(P,F ∩L ) = min{

i : L*(X +α,Y +β ) - Fi(X ,Y )}.

Now, the following result is a straightforward consequence of Definitions 1.1.10, 1.1.27,1.1.29 and 1.1.32.

Proposition 1.1.34. Let F be a plane curve and let P ∈F . If L is a line, then L is tangent toF at P if and only if L is the tangent line of some branch b of F centered at P.

Theorem 1.1.35. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 3.7). Let F :F(X ,Y,Z) = 0 and G : G(X ,Y,Z) = 0 be plane curves. If mF

P and mGP are the multiplicities of

F and G at P, respectively, then

I(P,F ∩G )>mFP mG

P , (1.1)

with equality occurring in (1.1) if and only if F and G have no common tangent at P.

Theorem 1.1.36 (Bézout’s Theorem). Let F : F(X ,Y,Z) = 0 and G : G(X ,Y,Z) = 0 be planecurves of degrees d1 and d2, respectively. If F and G do not have a common component, then

∑P∈F∩G

I(P,F ∩G ) = d1d2.

Proof. A proof of this result can be found in (HIRSCHFELD; KORCHMÁROS; TORRES,2008, Theorem 3.14). �

This subsection ends with the following definition, which is important in Part II.

Definition 1.1.37. Let F be a plane curve, let P be a nonsingular point of F , and consider TP

the tangent line to F at P. P is an inflection point of F if

I(P,F ∩TP)> 3.

Further, if d > 2, then P is called a total inflection point of F if it is the only point of F on thetangent line TP. In this case, I(P,F ∩TP) = d.

1.1.3 Function fields of irreducible plane curves

Definition 1.1.38. Let F : F(X ,Y,Z) = 0 be a plane curve. A generic point of F is a point(x : y : z) satisfying the following conditions:

1. The elements x, y and z are in some extension of K and satisfy F(x,y,z) = 0.

2. If G(X ,Y,Z) ∈ K[X ,Y,Z] is a homogeneous polynomial such that G(x,y,z) = 0, thenF(X ,Y,Z) | G(X ,Y,Z).

1.2. Algebraic function fields 35

Theorem 1.1.39. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 5.3). A planecurve has a generic point if and only if it is irreducible.

Example 1.1.40. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Corollary 5.8). Let F :F(X ,Y,Z) = 0 be an irreducible plane curve and let b be a branch of F . Then, all primitiverepresentations ă of b are generic points of F .

Definition 1.1.41. Let F : F(X ,Y,Z) = 0 be an irreducible plane curve and let (x : y : z) be ageneric point of F . The function field of F , denoted by K(F ), is the subfield of K(x,y,z){

A(x,y,z)B(x,y,z)

: A(X ,Y,Z),B(X ,Y,Z) are homogeneous polynomialsin K[X ,Y,Z] of the same degree, and B(x,y,z) =0

}.

Proposition 1.1.42. Let F : F(X ,Y,Z) = 0 be an irreducible plane curve. Then its function fieldK(F ) is uniquely determined up to K-isomorphisms.


Remark 1.1.43. Let F : F(X ,Y,Z) = 0 be an irreducible plane curve and let (x : y : z) be a

generic point of F . If z = 0, then K(F ) is K-isomorphic to K(

xz,yz

), where F(x/z,y/z,1) = 0.

A similar fact occurs if x = 0 or y = 0.

Definition 1.1.44. Two irreducible plane curves are birationally equivalent if their functionfields are K-isomorphic.

This subsection ends with the following property concerning the function field of anirreducible plane curve.

Proposition 1.1.45. Let F : F(X ,Y,Z) = 0 be an irreducible plane curve. Then K(F ) is anextension of transcendence degree 1 over K.


1.2 Algebraic function fields

In all this section, let Σ be a field of transcendence degree 1 over K. In general, Σ/Kis called an algebraic function field of one variable over K or simply a function field, and theelements of Σ∖K are called functions.


1.2.1 Plane models

Theorem 1.2.1 (Theorem of the Primitive Element). If x ∈ Σ is a transcendent element over K,then there exists y ∈ Σ such that Σ =K(x,y).

Proof. A proof of this result can be found in (HIRSCHFELD; KORCHMÁROS; TORRES,2008, Theorem A.5). �

Definition 1.2.2. A plane model of Σ is a pair (F ,(x,y)), where Σ=K(x,y) and F : F(X ,Y )= 0is an irreducible plane curve having (x : y : 1) as a generic point.

As a consequence of Remark 1.1.43, Definition 1.1.44 and Theorem 1.2.1, the followingoccurs.

Corollary 1.2.3. There exists a plane model of Σ. Further, if (F ,(x1,y1)) and (G ,(x2,y2)) aretwo plane models of Σ, then F and G are birationally equivalent.

1.2.2 Places

Let (F ,(x,y)) be a plane model of Σ.

Definition 1.2.4. 1. A place representation of Σ is a K-monomorphism τ : Σ→K((T )).

2. A place representation τ of Σ is primitive if ă = (τ(x) : τ(y) : 1) is a primitive branchrepresentation.

3. Two place representations τ1 and τ2 are equivalent if there exists a K-automorphism σ ofK((T )) such that τ1 = σ ∘ τ2.

4. A place P of Σ is an equivalence class of primitive place representations.

Theorem 1.2.5. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 5.28). The placesof Σ and the branches of F are in a natural one-to-one correspondence.

Definition 1.2.6. Let P be a place of Σ. The order of f ∈ Σ at P is the number

vP( f ) := ordT (τ( f )),

where τ is a primitive representation of P .

Definition 1.2.7. Let P be a place of Σ. A local parameter at P is an element t ∈ Σ such thatvP(t) = 1.

Proposition 1.2.8. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Lemma 5.31). Forevery place P of Σ there exists a local parameter at P .

Definition 1.2.9. Let f ∈ Σ and let P be a place of Σ.

1. If vP( f )> 0, then P is a zero of f of multiplicity vP( f ).


2. If vP( f )< 0, then P is a pole of f of multiplicity −vP( f ).

Theorem 1.2.10. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorems 5.33, 5.34 and5.35). Every function f ∈ Σ∖K has only finitely many zeros and poles. Further, the number ofzeros and the number of poles of f (counted with their multiplicities) are equal and given by[Σ : K( f )].

Definition 1.2.11. Let f ∈ Σ and let P be a place of Σ such that vP( f ) > 0. Then, one candefine f (P) := α , where α is the only element in K satisfying vP( f −α)> 0.

From the definitions, the following properties hold.

Proposition 1.2.12. Let P be a place of Σ and let f1, f2 ∈ Σ be such that vP( f1) > 0 andvP( f2)> 0. Then, ( f1 + f2)(P) = f1(P)+ f2(P) and ( f1 f2)(P) = f1(P) f2(P).

1.2.3 Some remarks

Definition 1.2.13. A valuation ring of Σ/K is a ring O satisfying the following properties:

1. K( O ( Σ.

2. For each f ∈ Σ, f ∈ O or f−1 ∈ O .

Definition 1.2.14. A discrete valuation of Σ/K is a surjective function v : Σ→Z∪{∞} satisfying:

1. v( f ) = ∞ if and only if f = 0.

2. v( f1 f2) = v( f1)+ v( f2), for all f1, f2 ∈ Σ.

3. v( f1 + f2)>min{v( f1),v( f2)}, for all f1, f2 ∈ Σ.

4. v(α) = 0, for all α ∈K.

Theorem 1.2.15. (STICHTENOTH, 2009, Theorem 1.1.13). Each valuation ring of Σ/K givesrise to a discrete valuation and vice-versa.

Considering Theorem 1.2.15, the following result establishes an important connectionbetween the definition of a place given in Subsection 1.2.2 and the usual one, which defines aplace of Σ as the maximal ideal of a valuation ring of Σ/K.

Theorem 1.2.16. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 7.137). Foreach place P of Σ, vP defines a discrete valuation of Σ/K. Conversely, every discrete valuationof Σ/K is of the form vP for some place P of Σ.


1.2.4 Expansion of a function at a local parameter

Let P be a place of Σ.

Definition 1.2.17. ΣP is a completion of Σ with respect to the discrete valuation vP if:

1. ΣP is a field extension of Σ.

2. ΣP has a discrete valuation vP that extends vP .

3. ΣP is complete with respect to vP , that is, every sequence ( fi)i>0 in ΣP satisfying theproperty that

for all k ∈ R there is an index i0 ∈ N such that vP( fi1− fi2)> k whenever i1, i2 > i0

is convergent in the following sense:

there exists f ∈ ΣP satisfying the condition that for all k ∈ R there is an index i0 ∈ Nsuch that vP( f − fi)> k whenever i> i0.

4. Σ is dense in ΣP , that is, for each f ∈ ΣP there is a sequence ( fi)i>0 in Σ such thatlimi→∞

fi = f .

Proposition 1.2.18. (STICHTENOTH, 2009, Proposition 4.2.3). There exists a completion ΣP

of Σ with respect to vP . Moreover, this completion is unique in the following sense: if Σ(1)P is

another completion of Σ with respect to vP , then there is a unique isomorphism between ΣP

and Σ(1)P such that the following diagram commutes

ΣP

vP ##

∼= Σ(1)P

v(1)P{{Z∪{∞}

where v(1)P is the discrete valuation of Σ(1)P that extends vP .

Based on Proposition 1.2.18, from now on ΣP denotes the completion of Σ with respectto the valuation vP .

Definition 1.2.19. A series∞

∑i=k

fi is convergent in ΣP if its sequence of partial sums is convergent.

With this definition, the following result holds.

Theorem 1.2.20. (STICHTENOTH, 2009, Theorem 4.2.6). Let t ∈ Σ be a local parameter at

P . Then, every element f ∈ ΣP has a unique representation of the form f =∞

∑i=k

αiti, with k ∈ Z

and αi ∈K. Conversely, if (αi)i>k is a sequence in K, then the series∞

∑i=k

αiti is convergent in ΣP

and vP

(∞

∑i=k

αiti)= min

{i : αi = 0

}.


Now, let τ be a primitive representation of P and let t ∈ Σ be a local parameter at P .Since ordT (τ(t)) = 1, given f ∈ Σ∖{0}, one may write

τ( f ) =∞

∑i=vP ( f )

βiτ(t)i,

with βi ∈ K. Further, for any other primitive representation τ1 of P , τ1 = σ ∘ τ , for someK-automorphism σ of K((T )), and then

τ1( f ) = σ ∘ τ( f )1=

∞

∑i=vP ( f )

βi(σ ∘ τ)(t)i =∞

∑i=vP ( f )

βiτ1(t)i,

which shows that the coefficients βi are uniquely determined by P and t.

As a consequence of Theorem 1.2.20, especially of its proof, the following equalityoccurs:

f =∞

∑i=vP ( f )

βiti ∈ ΣP .

Definition 1.2.21. The representation f :=∞

∑i=vP ( f )

βiti ∈ ΣP is the P-adic expansion of f at

the local parameter t at P .

1.2.5 Separating variables

Definition 1.2.22. A separating variable of Σ is an element x∈ Σ such that the extension Σ/K(x)

is separable.

Proposition 1.2.23. If Σ =K(x,y), then either x or y is a separating variable of Σ.

Proof. The result follows from (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Lemma5.38). �

Proposition 1.2.24. Let P be a place of Σ. Then, every local parameter t at P is a separatingvariable.

Proof. The result follows from (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Lemma5.38). �

1.2.6 Derivatives

Let x ∈ Σ be a separating variable of Σ.

1 An idea for the proof of this equality can be found in (HIRSCHFELD; KORCHMÁROS; TORRES,2008, Considerations on Page 67).


Definition 1.2.25. The formal derivative in K[x] is the K-linear mapping

ddx

: K[x] → K[x]

F(x) =l

∑k=0

αkxk ↦→ dF(x)dx

:=l

∑k=1

kαkxk−1.

A straightforward verification shows the following result.

Proposition 1.2.26. The formal derivative in K[x] is a derivation of K[x] into itself, that is, itsatisfies the product rule:

dF(x)G(x)dx

= F(x)dG(x)

dx+G(x)

dF(x)dx

for each F(x),G(x) ∈ K[x]. Further,ddx

is over K, that is,dα

dx= 0, for all α ∈ K, and it has a

unique extension to a derivation of K(x) into itself, which satisfies

dF(x)/G(x)dx

=G(x)dF(x)

dx −F(x)dG(x)dx

G(x)2 ,

for each F(x),G(x) ∈K[x], with G(x) = 0.

Since x is a separating variable of Σ, and then Σ/K(x) is a separable extension, thefollowing occurs.

Proposition 1.2.27. (STICHTENOTH, 2009, Proposition 4.1.4). There exists a unique extension

ofddx

to a derivation of Σ into itself.

Remark 1.2.28. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Definition 5.47). For each

f ∈ Σ, let F(X ,Y ) ∈K[X ,Y ] be an irreducible polynomial such that F(x, f ) = 0. Then,d fdx

canbe explicitly defined by

d fdx

:=−∂F∂X |(x, f )∂F∂Y |(x, f )

.

Finally, this subsection ends with the following definition.

Definition 1.2.29. Let f ∈ Σ be an arbitrary element. Then, it is possible to define

d0 fdx0 := f ,

d1 fdx1 :=

d fdx

and, for each i> 2,

di fdxi :=

d(di−1 f/dxi−1)

dx,

which is called the i-th higher derivative of f ∈ Σ.


1.2.7 Hasse derivatives

Let x ∈ Σ be a separating variable of Σ.

Definition 1.2.30. For i,k ∈ N, define

D(i)x (xk) :=

(ki

)xk−i.

The i-th Hasse derivative on K[x] is the K-linear extension of D(i)x to K[x].

From Definition 1.2.30, the following holds.

Proposition 1.2.31. The mappings D(i)x on K[x] satisfy:

D(i)x ( f1 f2) =

i

∑k=0

D(k)x ( f1)D

(i−k)x ( f2), for all f1, f2 ∈K[x].

Proposition 1.2.32. The mappings D(i)x on K[x] extend uniquely to K-linear mappings of K(x)

into itself. These mappings are called the i-th Hasse derivative on K(x) and satisfy

D(i)x ( f1 f2) =

i

∑k=0

D(k)x ( f1)D

(i−k)x ( f2), for all f1, f2 ∈K(x).

Proof. The result follows from (GOLDSCHMIDT, 2003, Lemma 1.3.9). �

Since x is a separating variable of Σ, and then Σ/K(x) is a separable extension, thefollowing occurs.

Proposition 1.2.33. (GOLDSCHMIDT, 2003, Theorem 1.3.11). The mappings D(i)x on K(x)

extend uniquely to K-linear mappings of Σ into itself. These mappings are called the i-th Hasse

derivative on Σ and satisfy

D(i)x ( f1 f2) =

i

∑k=0

D(k)x ( f1)D

(i−k)x ( f2), for all f1, f2 ∈ Σ.

Proposition 1.2.34. (GOLDSCHMIDT, 2003, Lemma 1.3.13). If p = 0 or if i < p, then

D(i)x ( f ) =

1i!

di fdxi

for each f ∈ Σ.

Remark 1.2.35. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Definition 5.78). Letf ∈ Σ and let F(X ,Y ) ∈K[X ,Y ] be an irreducible polynomial satisfying F(x, f ) = 0. One can


define D(i)x ( f ) recursively using F(X ,Y ). More precisely, for i ∈N, the i-th Hasse derivative D(i)

x

of f ∈ Σ can be defined by D(0)x ( f ) := f and for i> 1

D(i)x ( f ) := − 1

∂F∂Y

∣∣∣∣(x, f )

[∂ (i)F∂X (i)

∣∣∣∣(x, f )

+i−1

∑k=1

∂ (i−k+1)F∂X (i−k)∂Y

∣∣∣∣(x, f )

D(k)x ( f )

+i

∑k=2

i

∑l=k

∑i1+···+ik=l

∂ (i−l+k)F∂X (i−l)∂Y (k)

∣∣∣∣(x, f )

D(i1)x ( f ) · · ·D(ik)

x ( f )],

where writing F(X ,Y ) = ∑αk1,k2Xk1Y k2 ,

∂ (i1+i2)F∂X (i1)∂Y (i2)

:= ∑αk1,k2

(k1

i1

)(k2

i2

)Xk1−i1Y k2−i2 .

Proposition 1.2.36. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Lemma 5.80 andTheorem 5.82). The i-th Hasse derivative on Σ, as established in Proposition 1.2.33 and Remark1.2.35, satisfy:

1. D(i1)x ∘D(i2)

x =

(i1 + i2

i1

)D(i1+i2)

x , for all i1, i2 ∈ N.

2. If x1 ∈ Σ is another separating variable, then, for each f ∈ Σ,

D(i)x1 ( f ) =

(dxdx1

)i

D(i)x ( f )+

i−1

∑k=1

fkD(k)x ( f ),

where f1, . . . , fi−1 ∈ Σ are polynomials in the indeterminates D(k)x1 (x), for 16 k 6 i.

Let P be a place of Σ and consider t a local parameter at P . This subsection endswith the following result which associates the P-adic expansions of f and D(i)

t ( f ) at the localparameter t at P , for each f ∈ Σ and i ∈ N.

Theorem 1.2.37. (GOLDSCHMIDT, 2003, Theorem 2.5.13). If f =∞

∑k=vP ( f )

αktk is the P-adic

expansion of f ∈Σ at the local parameter t at P , then, for each i∈N, D(i)t ( f ) =

∞

∑k=vP ( f )

(ki

)αkt

k−i

is the P-adic expansion of D(i)t ( f ) at the local parameter t at P .

1.2.8 Differentials and the notion of genus

Definition 1.2.38. Let x be a separating variable of Σ.

1. A differential is an element f dx ∈ Σ(dx), where f ∈ Σ and Σ(dx) is a transcendentalextension of Σ by the symbol dx.

2. For f ∈ Σ, the differential of f , denoted by d f , is the differential

d fdx

dx.


Remark 1.2.39. Let x,x1 ∈ Σ be two separating variables. Identifying dx1 with dx1dx dx, then, from

(HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 5.49), for each f ∈ Σ it followsthat

d fdx1

dx1 =d fdx1· dx1

dxdx =

d fdx

dx.

In other words, the definition of d f does not depend (in the previous sense) on the separatingvariable considered.

Definition 1.2.40. Let x ∈ Σ be a separating variable. For each place P of Σ, the order vP(dx)

of dx at P is defined by

vP(dx) := ordTdτ(x)

dT,

where τ is a primitive representation of P . In this case, P is a zero (resp. a pole) of dx ifvP(dx)> 0 (resp. vP(dx)< 0).

Proposition 1.2.41. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Lemma 5.53). If x∈ Σ

is a separating variable, then vP(dx) = 0 for almost all places P of Σ.

Proposition 1.2.42. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Lemma 5.54). Letx,x1 be two separating variables of Σ. Then

∑P is a place of Σ

vP(dx) = ∑P is a place of Σ

vP(dx1).

Definition 1.2.43. The genus g of Σ is the non-negative integer defined by the following equation


vP(dx) = 2g−2,

where x ∈ Σ is a separating variable.

Definition 1.2.44. The genus of an irreducible plane curve F is the genus of its function fieldK(F ).

Theorem 1.2.45. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 5.57 and Com-ments on Page 135). Let F be an irreducible plane curve of genus g and degree d.

1. If F is nonsingular, then g = (d−1)(d−2)2 .

2. If F is a singular curve and {P1, . . . ,Pk} is the set of its singularities, then

g6(d−1)(d−2)

2−

k

∑i=1

mPi(mPi−1)2

, (1.2)

where, for each i ∈ {1, . . . ,k}, mPi is the multiplicity of F at Pi. Moreover, if all thesingularities of F are ordinary, then the equality in (1.2) holds.


1.2.9 Divisors and linear series

Definition 1.2.46. The group of divisors of Σ, denoted by Div(Σ), is the free abelian groupgenerated by the places of Σ. Its elements are called divisors of Σ.

Definition 1.2.47. Let D = ∑P is a place of Σ

nPP ∈ Div(Σ).

1. The multiplicity of a place P in D is the number vP(D) := nP .

2. The support of D is the finite set

Supp(D) :={

P is a place of Σ : vP(D) = 0}.

Remark 1.2.48. The relation

D1 > D2 if and only if vP(D1)> vP(D2) for all places P of Σ

is a partial ordering on Div(Σ).

Definition 1.2.49. If D > 0, then D is an effective divisor. Otherwise, D is called a virtual

divisor.

Definition 1.2.50. The degree of D ∈ Div(Σ) is defined by

deg(D) := ∑P is a place of Σ

vP(D).

Remark 1.2.51. The mapping

deg : Div(Σ) → ZD ↦→ deg(D)

is a homomorphism of groups.

From Theorem 1.2.10, the following definition is obtained.

Definition 1.2.52. Let f ∈ Σ∖{0}.

1. The zero divisor of f is the divisor div( f )0 := ∑P is a place of Σ

vP ( f )>0

vP( f )P .

2. The pole divisor of f is the divisor div( f )∞ := ∑P is a place of Σ

vP ( f )<0

−vP( f )P .

3. The principal divisor of f is the divisor div( f ) := div( f )0−div( f )∞.

Definition 1.2.53. Two divisors D1 and D2 of Σ are equivalent if D1 = div( f )+D2, for somef ∈ Σ∖{0}. In this case, the relation between D1 and D2 is denoted by D1 ∼ D2.

Remark 1.2.54. The relation established in Definition 1.2.53 is an equivalence relation.


The following property follows immediately from the definitions.

Proposition 1.2.55. If D1 ∼ D2, then deg(D1) = deg(D2).

Definition 1.2.56. Let x ∈ Σ be a separating variable.

1. The canonical divisor associated with the differential dx is defined by

div(dx) := ∑P is a place of Σ

vP(dx).

2. For f ∈ Σ∖{0}, the canonical divisor associated with the differential f dx is defined by

div( f dx) := div( f )+div(dx).

In particular, by definition, the divisors div( f dx), with f ∈ Σ∖{0}, constitute a class of divisors,which is called the canonical class.

Remark 1.2.57. In terms of the identification presented in Remark 1.2.39, the definition of thecanonical class does not depend on the separating variable considered.

Definition 1.2.58. For each D ∈ Div(Σ), the Riemann-Roch space associated to D is the vectorspace over K defined by

L (D) :={

f ∈ Σ∖{0} : div( f )+D> 0}∪{

0}.

Notation 1.2.59. For each D ∈ Div(Σ), the dimension of L (D) over K is denoted by `(D).

Proposition 1.2.60. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Lemma 6.69). Let D,D1 and D2 be divisors of Σ.

1. If D > 0, then `(D) 6 deg(D)+ 1. More generally, if D2 > D1, then `(D2)− `(D1) 6

deg(D2)−deg(D1).

2. If D1 ∼ D2, then L (D1) and L (D2) are K-isomorphic.

Theorem 1.2.61 (Riemann-Roch Theorem). Let x ∈ Σ be a separating variable. For each D ∈Div(Σ),

`(D) = deg(D)−g+1+ `(div(dx)−D),

where g is the genus of Σ.


Definition 1.2.62. For any place P of Σ, a non-negative integer i is a pole number of P if thereis a function f ∈ Σ such that div( f )∞ = iP . Otherwise, i is called a gap number of P .


Definition 1.2.63. Associated to a divisor E ∈ Div(Σ), it is possible to define the following set:

|E| :={

div( f )+E : f ∈L (E)∖{0}}.

Remark 1.2.64. Let E ∈ Div(Σ). Then, the mapping

div( f )+E ∈ |E| ↦→ [ f ] ∈ P(L (E))

provides a structure of projective space on |E|.

Notation 1.2.65. For E ∈ Div(Σ), the following notation is used: |E| ∼= P(L (E)).

Definition 1.2.66. A linear series D on Σ is a subset of some |E| of the form{div( f )+E : f ∈S ∖{0}

},

where S is a K-linear subspace of L (E). The numbers d := deg(E) and N = dim(D) :=dim(S )−1 are, respectively, the degree and the dimension of D , and D is said to be a gN

d on Σ.Further, D is called complete if D = |E|.

Notation 1.2.67. In terms of Definition 1.2.66, for a linear series D the following notation isused: D ∼= P(S )⊆ |E|.

Definition 1.2.68. Let x ∈ Σ be a separating variable and let g > 1 be the genus of Σ. Thecomplete linear series |div(dx)| of degree 2g− 2 and dimension g− 1 is the canonical linear

series.

Definition 1.2.69. The linear series D1 ∼= P(S1)⊆ |E1| is a subseries of D2 ∼= P(S2)⊆ |E2| ifL (E1)⊆L (E2) and S1 ⊆S2.

1.2.10 Linear systems of plane curves

Definition 1.2.70. For each i = 0, . . . ,M, let Fi : Fi(X ,Y,Z) = 0 be a plane curve of degree d,with F0(X ,Y,Z), . . . ,FM(X ,Y,Z) linearly independent over K. Then, the set{

F :M

∑i=0

αiFi(X ,Y,Z) = 0 : (α0 : · · · : αM) ∈ PM(K)

}is the linear system of degree d and dimension M generated by F0, . . . ,FM.

Definition 1.2.71. Let (F ,(x,y)) be a plane model of Σ. If G is a plane curve not containing F

as a component, then the intersection divisor cut out on F by G is defined by

G ∙F = ∑P is a place of Σ

I(P,G ∩b)P,

where, for each place P of Σ, b is the corresponding branch of F and P ∈F is its center.

1.3. Space curves 47

Theorem 1.2.72. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 6.46). Let(F ,(x,y)) be a plane model of Σ. The plane curves of a linear system of degree d that not containF as a component cut out on F the divisors of a linear series. Further, the converse holds, up toa fixed divisor.

Proposition 1.2.73. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Remark 6.47 andComments on Page 172). Let (F ,(x,y)) be a plane model of Σ of degree d. If D is the linearseries cut out on F by the plane curves (not containing F as a component) of a linear system ofdegree d< d, then the dimension of D is equal to the dimension of the linear system.

1.3 Space curvesLet M be a positive integer.

Definition 1.3.1. A branch representation ă is a point of PM(K((T )))∖PM(K).

All definitions and results presented in Subsection 1.1.2 remain true for a generalM. In particular, the notions of special coordinates, center of a branch representation, affine

(special) coordinates, equivalent branch representations, (im)primitive branch representations,and ramification index still hold, and also:

Definition 1.3.2. A branch b is an equivalence class of primitive branch representations. Thecenter of a branch is the center of any of its primitive representations.

Definition 1.3.3. A hypersurface ∆⊆ PM(K) is the point set given by the solutions of

F(X0, . . . ,XM) = 0

in PM(K), for some homogeneous polynomial F(X0, . . . ,XM) ∈K[X0, . . . ,XM] of degree d > 1.In this case, d is the degree of ∆. Further, if d = 1, then ∆ is called a hyperplane.

Definition 1.3.4. Let ∆ : F(X0, . . . ,XM) = 0 be a hypersurface and let b be a branch centered atthe point Q. If (α0(T ) : · · · : αM(T )) are special coordinates for a primitive representation ă of b,then the intersection multiplicity of ∆ and b is defined by

I(Q,∆∩b) := ordT (F(α0(T ), . . . , αM(T ))).

Definition 1.3.5. Let b be a branch centered at the point Q. Then b is linear if I(Q,∆∩b) = 1for some hyperplane ∆.

Now, let Σ be a field of transcendence degree 1 over K and let (x0 : · · · : xM) ∈ PM(Σ) besuch that

Σ =K(

x0

xk, . . . ,

xM

xk

)


for some k ∈ {0, . . . ,M} satisfying xk = 0. Note that if xl = 0 for another l ∈ {0, . . . ,M}, thenthe following equality also occurs:

Σ =K(

x0

xl, . . . ,

xM

xl

).

Proposition 1.3.6. There exists i ∈ {0, . . . ,M} such thatxi

xkis a separating variable of Σ.

Proof. The result follows from Proposition 1.2.23 and (HIRSCHFELD; KORCHMÁROS;TORRES, 2008, Lemma 5.38). �

For each place P of Σ, let τ be a primitive representation of P . Then,

ă = (τ(x0) : · · · : τ(xM))

is a primitive branch representation.

Definition 1.3.7. The branch b with primitive representation given by ă = (τ(x0) : · · · : τ(xM))

is the branch associated to the place P with respect to (x0 : · · · : xM) ∈ PM(Σ).

Based on this, the notion of a projective irreducible algebraic curve can be established.

Definition 1.3.8. A projective irreducible algebraic curve X in PM(K) is a set of the form{(α0 : · · · : αM) ∈ PM(K) : (α0 : ··· :αM) is the center of a branch associated to a place

P of Σ with respect to (x0 : ··· :xM)∈PM(Σ)

},

where Σ is a field of transcendence degree 1 over K and (x0 : · · · : xM) ∈ PM(Σ) is a fixed elementsuch that

Σ =K(

x0

xk, . . . ,

xM

xk

)for some k ∈ {0, . . . ,M} satisfying xk = 0. In this case, K(X ) := Σ is the function field of X ,the branches associated to the places of Σ with respect to (x0 : · · · : xM) ∈ PM(Σ) are the branches

of X , the genus of X is the genus of K(X ), and

(X ,(x0 : · · · : xM))

is a model of Σ in PM(K). Further, x0, . . . ,xM are called the coordinate functions of X .

Notation 1.3.9. Hereafter, the expression irreducible curve means a projective irreduciblealgebraic curve. Further, if Σ is a field of transcendence degree 1 over K, (x0 : · · · : xM) ∈ PM(Σ)

is a fixed element such that

Σ =K(

x0

xk, . . . ,

xM

xk

)


for some k ∈ {0, . . . ,M} satisfying xk = 0, and

X =

{(α0 : · · · : αM) ∈ PM(K) : (α0 : ··· :αM) is the center of a branch associated to a place

P of Σ with respect to (x0 : ··· :xM)∈PM(Σ)

},

then X is sometimes called the irreducible curve given by (x0 : · · · : xM), without mention thefunction field Σ.

Theorem 1.3.10. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 7.2). Let X

be an irreducible curve. Then each point of X is the center of a finite number of branches of X .


be an irreducible curve. Then X has an infinite number of points.


be an irreducible curve given by the point (x0 : · · · : xM).

1. The hypersurface ∆ : F(X0, . . . ,XM) = 0 contains X if and only if F(x0, . . . ,xM) = 0.

2. If ∆ is a hypersurface not containing X , then ∆∩X is a finite set.

Theorem 1.3.13. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 7.6). Let b bea branch centered at a point Q of an irreducible curve X , and let (α0(T ), . . . , αM(T )) be specialcoordinates for a primitive representation ă of b. If ∆ : F(X0, . . . ,XM) = 0 is a hypersurface notcontaining X , then I(Q,∆∩b) is finite, that is, F(α0(T ), . . . , αM(T )) = 0.

Definition 1.3.14. Let X be an irreducible curve and let ∆ be a hypersurface not containing X .The intersection divisor of X and ∆ is defined by

∆∙X := ∑b is a branch of X centered at Q

and associated to the place P of K(X )

I(Q,∆∩b)P.


be an irreducible curve given by the point (x0 : · · · : xM) and let ∆ : F(X0, . . . ,XM) = 0 be ahypersurface of degree d not containing X . If ∆i : Xi = 0 does not contain X , that is, if xi = 0,then

∆∙X = div(F(x0/xi, . . . ,xM/xi))+d(∆i ∙X ).

Definition 1.3.16. Let X be an irreducible curve. If ∆ is a hyperplane not containing X , thenthe degree of X is defined by

deg(X ) := deg(∆∙X ).

Corollary 1.3.17. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Corollary 7.9). Let X

be an irreducible curve given by the point (x0 : · · · : xM) and let ∆ : F(X0, . . . ,XM) = 0 be ahypersurface of degree d not containing X . Then

deg(∆∙X ) = d ·deg(X ).


Definition 1.3.18. Let X be an irreducible curve. A point Q of X is singular if it is either thecenter of at least two branches or the center of one nonlinear branch of X . Otherwise, Q iscalled nonsingular.

This subsection ends with the following result regarding the number of singularities ofan irreducible curve.


be an irreducible curve. Then X has a finite number of singular points.

1.3.1 Rational transformations and morphisms

Let Σ be a field of transcendence degree 1 over K and let (x0 : · · · : xM) ∈ PM(Σ) be suchthat

Σ =K(

x0

xi, . . . ,

xM

xi

)for some i ∈ {0, . . . ,M}. Also, let X be the irreducible curve given by (x0 : · · · : xM).

Definition 1.3.20. A rational transformation φ on X is an element φ := ( f0 : · · · : fN) ∈ PN(Σ).In this case, f0, . . . , fN ∈ Σ are called the coordinates of φ .

Definition 1.3.21. Let φ = ( f0 : · · · : fN) ∈ PN(Σ) be a rational transformation on X and letQ = (α0 : · · · : αM) ∈X . φ is defined at Q if it is possible to write

( f0 : · · · : fN) = (A0(x0, . . . ,xM) : · · · : AN(x0, . . . ,xM)) ∈ PN(Σ),

where A0(X0, . . . ,XM), . . . ,AN(X0, . . . ,XM)∈K[X0, . . . ,XM] are homogeneous polynomials of thesame degree, and Ak(α0, . . . ,αM) = 0 for some k ∈ {0, . . . ,N}. In this case,

φ(Q) := (A0(α0, . . . ,αM) : · · · : AN(α0, . . . ,αM)).

Notation 1.3.22. A rational transformation φ = ( f0 : · · · : fN) ∈ PN(Σ) on X is denoted by

φ = ( f0 : · · · : fN) : X 99K PN(K).

Definition 1.3.23. A rational transformation φ = ( f0 : · · · : fN) : X 99K PN(K) is a morphism

if it is defined at every point of X . In this case, the notation φ = ( f0 : · · · : fN) : X → PN(K) isused.

Remark 1.3.24. Let φ = ( f0 : · · · : fN) : X 99K PN(K) be a rational transformation on X andlet

Σ′ =K

(f0

fl, . . . ,

fN

fl

)⊆ Σ,

where l ∈ {0, . . . ,N} is such that fl = 0. Then one of the following situations occurs:


1. K=Σ′: in this case, φ =( f0 : · · · : fN)∈PN(K) and, in particular, φ is a constant morphism.

2. K( Σ′: in this case, Σ′ is also a field of transcendence degree 1 over K and so it is possibleto define the irreducible curve Y given by the point ( f0/ fl : · · · : fN/ fl) ∈ PN(Σ′)

Y :={(β0 : · · · : βN) ∈ PN(K) : (β0 : ··· :βN) is the center of a branch associated to a place

P ′ of Σ′ with respect to ( f0/ fl : ··· : fN/ fl)∈PN(Σ′)

}.

Proposition 1.3.25. Considering the notation as in Remark 1.3.24, if Σ′ is a field of transcendencedegree 1 over K and φ is defined at Q ∈X , then φ(Q) ∈ Y .

Proof. The result follows from (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Considera-tions on Section 7.2). �

Definition 1.3.26. Consider the notation as in Remark 1.3.24.

1. If Σ′ is a field of transcendence degree 1 over K, then Y is the image curve of φ anddeg(φ) := [Σ : Σ′] is the degree of φ . Further, in this case the notation φ = ( f0 : · · · : fN) :X 99K Y is used, and φ is called a covering.

2. If Σ′ = Σ, then X and Y are said to be birationally equivalent and φ is a birational trans-

formation. In this case, there exists a rational transformation φ−1 : Y 99KX satisfyingφ−1 ∘φ = IdX and φ ∘φ−1 = IdY , where φ−1 ∘φ = IdX means that, for each Q ∈X

such that φ is defined at Q and φ−1 is defined at φ(Q), the equality φ−1(φ(Q)) = Q holds,and similarly for φ ∘φ−1 = IdY .

3. If Σ′ = Σ and φ is a morphism, then φ is called a birational morphism. Further, if φ−1 isalso a morphism, then φ is called an isomorphism.

Theorem 1.3.27. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 7.17). Everyirreducible curve is birationally equivalent to an irreducible plane curve.

Definition 1.3.28. Two arbitrary irreducible curves X and Y are birationally equivalent if theirfunction fields are K-isomorphic.

Proposition 1.3.29. Let φ = ( f0 : · · · : fN) : X 99K PN(K) be a rational transformation on X .If Q ∈X is the center of only one branch of X , then φ is defined at Q. Moreover, φ(Q) can bedetermined as follows: let P be the unique place of Σ corresponding to Q ∈X , let

eP =−min{

vP( f0), . . . ,vP( fN)

},

and let t ∈ Σ be a local parameter at P . Then, vP(teP fi) > 0, for all i = 0, . . . ,N, and

vP(teP fi) = 0 for at least one i = 0, . . . ,N. Therefore,

φ(Q) = ((teP f0)(P) : · · · : (teP fN)(P)) ∈ PN(K),

where (teP fi)(P) is given by Definition 1.2.11, for each i ∈ {0, . . . ,N}.


Proof. The result follows from (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Considera-tions on Section 7.2). �

As a consequence of Proposition 1.3.29, the following result holds.

Corollary 1.3.30. Let φ = ( f0 : · · · : fN) : X 99K PN(K) be a rational transformation on X . IfX is nonsingular, then φ is a morphism.

Definition 1.3.31. Let φ = ( f0 : · · · : fN) : X → PN(K) be a morphism. Then, φ is callednondegenerate if its image is not contained in a hyperplane of PN(K), which occurs by Theorem1.3.12 if and only if f0, . . . , fN are linearly independent over K.

1.3.2 Some remarks

In general, an irreducible curve is defined as a projective algebraic variety of dimension

1, being this concept equivalent to that one presented in Definition 1.3.8. More details on thissubject can be found in (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Section 7.17).

Now, let Σ be a field of transcendence degree 1 over K and let (X ,(x0 : · · · : xM)) be amodel of Σ in PM(K). Using the notion of projective algebraic varieties, the following resultestablishes the existence of a nonsingular model of Σ.

Theorem 1.3.32. (FULTON, 2008, Theorem 3 in Chapter 7). There exist a nonsingular irre-ducible curve X1 and a birational morphism φ1 : X1→X . Further, if X2 is another nonsingularirreducible curve for which there exists a birational morphism φ2 : X2→X , then there is aunique isomorphism φ : X1→X2 such that the following diagram commutes

X1

φ1 !!

φ−→ X2

φ2}}X

Definition 1.3.33. The nonsingular irreducible curve X1 given in Theorem 1.3.32 is the nonsin-

gular model of X .

Finally, from the definitions, the following property says that the notation introducedbelow is well defined.

Proposition 1.3.34. Suppose that X is a nonsingular irreducible curve. Then, there exist thefollowing one-to-one correspondences:

Places of Σ1−1←→ Branches of X

1−1←→ Points of X .

Notation 1.3.35. From now on, whenever X is considered a nonsingular irreducible curve, themain structures involving the places of Σ =K(X ) are expressed in terms of the corresponding


points of X . For instance, if Q ∈X is the point corresponding to the place P of Σ, thenthe expressions vQ, Div(X ) and ∑

Q∈XnQQ ∈ Div(X ) are used instead of vP , Div(Σ) and


nPP ∈ Div(Σ), respectively.

1.3.3 Morphisms × linear series

Let X be a nonsingular irreducible curve and let K(X ) be its function field.

1.3.3.1 Morphisms from linear series

Let D ∼= P(S )⊆ |E| be a linear series of dimension N and degree d= deg(E) on X .

Definition 1.3.36. For each Q ∈X and i ∈ N, define

Di(Q) :={

D ∈D : D> iQ}.

Proposition 1.3.37. (TORRES, 2000, Lemma 1.3). Let Q ∈X . For each i ∈ N, the followingholds.

1. Di(Q) is a subseries of D , namely Di(Q)∼= P(S ∩L (E− iQ))⊆ |E|.

2. Di+1(Q)⊆Di(Q) and Di(Q) = /0 if i > d.

3. dim(Di(Q))6 dim(Di+1(Q))+1.

Definition 1.3.38. Let Q ∈X . The multiplicity b(Q) of D at Q is defined by

b(Q) := min{

vQ(D) : D ∈D

}.

From the definitions, the following occurs.

Proposition 1.3.39. Let Q ∈X . Then b(Q)> 0 if and only if Q ∈ Supp(D) for all D ∈D . Inparticular, b(Q) = 0 for almost all points Q ∈X .

Definition 1.3.40. The base locus of D is the divisor B of X given by

B := ∑Q∈X

b(Q)Q.

If B = 0, then D is called a base-point-free linear series.

Proposition 1.3.41. (TORRES, 2000, Considerations on Page 7). Let B be the base locus of D .Then

DB :={

D−B : D ∈D

}is a base-point-free subseries of D of dimension N and degree d−deg(B), namely

DB ∼= P(S )⊆ |E−B|.


Proposition 1.3.42. (TORRES, 2000, Lemma 1.4). Let { f0, . . . , fM} be a set of generators forS . Then E is determined by D , that is, for each Q ∈X

vQ(E) = b(Q)−min{

vQ( f0), . . . ,vQ( fM)

}.

Now, a morphism φ is associated to D . In a coordinate-invariant description, sinceD = Db(Q)(Q)) Db(Q)+1(Q) and thus dim(Db(Q)+1(Q)) = d−1 by Proposition 1.3.37, define

φ : X → D* ∼= P(S )*

Q ↦→ Db(Q)+1(Q).

In order to provide coordinates for φ , let { f0, . . . , fN} be a basis for S . If t is a local parameterat Q, then, for each f ∈S ,

div( f )+E ∈Db(Q)+1(Q) ⇔ vQ(tvQ(E)−b(Q) f )> 1

⇔ (tvQ(E)−b(Q) f )(Q) = 0.

Therefore, writing f =N

∑k=0

αk fk, with (α0 : · · · : αN) ∈ PN(K),

Db(Q)+1(Q) ∼={(α0 : · · · : αN) ∈ PN(K) :

N

∑k=0

(tvQ(E)−b(Q) fk)(Q)αk = 0}∈ PN(K)*

∼= ((tvQ(E)−b(Q) f0)(Q) : · · · : (tvQ(E)−b(Q) fN)(Q)) ∈ PN(K),

which leads to define

φ := ( f0 : · · · : fN) : X → PN(K).

Formally:

Proposition 1.3.43. (TORRES, 2000, Lemma 1.5). Each basis { f0, . . . , fN} of S over K de-fines a nondegenerate morphism φ = ( f0 : · · · : fN) : X → PN(K). Moreover, a nondegeneratemorphism associated to D is uniquely determined by D up to a projective equivalence.

Remark 1.3.44. From the previous discussion, the morphisms that come from D and DB havethe same coordinate description.

This subsection ends with the definition and some properties of the canonical curve.

Definition 1.3.45. Let x ∈ K(X ) be a separating variable and let |div(dx)| be the canonicallinear series on X . The canonical curve is the irreducible image curve of the morphism thatcomes from |div(dx)|.

Theorem 1.3.46. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 7.45). Thecanonical curve is a nonsingular irreducible curve in Pg−1(K), where g is the genus of X .


1.3.3.2 Linear series from morphisms

Let φ = ( f0 : · · · : fN) : X → PN(K) be a morphism on X and, for each Q ∈X , let

eQ =−min{

vQ( f0), . . . ,vQ( fN)

}.

Since eQ = 0 for almost all points Q ∈X , then the divisor

E = ∑Q∈X

eQQ

is well defined. Further:

Proposition 1.3.47. For each i ∈ {0, . . . ,N}, fi ∈L (E).

Therefore, defining S as the K-linear subspace of L (E) generated by { f0, . . . , fN}, it ispossible to consider the base-point-free linear series on X

D =

{div( f )+E : f ∈S

}⊆ |E|.

More precisely, the following holds.

Proposition 1.3.48. (TORRES, 2000, Lemma 1.9). Associated to a morphism

φ = ( f0 : · · · : fN) : X → PN(K)

there exists a base-point-free linear series D ∼= P(S )⊆ |E|, where S is the K-linear subspaceof L (E) generated by { f0, . . . , fN}, and E is defined by

vQ(E) =−min{

vQ( f0), . . . ,vQ( fN)

}.

The linear series D is uniquely determined by φ and it is invariant under projective equivalenceof morphisms. Further, if φ is nondegenerate, then dim(D) = N.

Now, suppose that φ = ( f0 : · · · : fN) : X → PN(K) is a nondegenerate morphism onX . Then, as before, the corresponding linear series D is given by

D =

{div( N

∑i=0

αi fi

)+E : (α0 : · · · : αN) ∈ PN(K)

},

and since (α0 : · · · : αN) can be identified with the hyperplane

∆ :N

∑i=0

αiXi = 0,

it is possible to write

D =

{φ*(∆) : ∆ is a hyperplane in PN(K)

},

where φ*(∆) := div

( N

∑i=0

αi fi

)+E is the pullback of ∆ by φ .


Proposition 1.3.49. (TORRES, 2000, Lemma 1.11). Considering the previous notation:

1. Q ∈ Supp(φ*(∆)) if and only if φ(Q) ∈ ∆. Therefore, Supp(φ*(∆)) = φ−1(φ(X )∩∆),where φ(X ) is the image curve of φ .

2. deg(D) = deg(φ)deg(φ(X )). In particular, if φ is a birational morphism, then deg(D) =

deg(φ(X )).

3. If φ is a birational morphism, then, identifying the branches of φ(X ) with the points ofX , ∆∙φ(X ) = φ*(∆) for each hyperplane ∆.

57

CHAPTER

2PROJECTIVE ALGEBRAIC CURVES

DEFINED OVER FINITE FIELDS

Based on (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Chapters 8 and 9) and(GOLDSCHMIDT, 2003, Chapter 3), the purpose of this chapter is to provide some backgroundon the topic projective algebraic curves defined over finite fields. As in the previous chapter, inmany cases the objects introduced in the text are assumed to be well defined without proofs orother comments in this direction.

Here, let Fq be the finite field with q = pm elements, where p is a prime number, andconsider Fq the algebraic closure of Fq.

2.1 Plane curves defined over Fq

Definition 2.1.1. A plane curve F : F(X ,Y,Z) = 0 in P2(Fq) is defined over Fq if there existsλ ∈ Fq satisfying λF(X ,Y,Z) ∈ Fq[X ,Y,Z].

Proposition 2.1.2. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Lemma 8.2). Let F :F(X ,Y,Z) = 0 be an irreducible plane curve in P2(Fq). Then, F is defined over Fq if and onlyif (αq : β q : γq) ∈F , for all (α : β : γ) ∈F .

Proposition 2.1.3. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Lemma 8.3). Let F :F(X ,Y,Z) = 0 be an irreducible plane curve in P2(Fq). Then, F is defined over Fq if and onlyif, for every generic point (x : y : z) of F , (xq : yq : zq) is also a generic point of F .

Definition 2.1.4. Let F : F(X ,Y,Z) = 0 be a plane curve in P2(Fq) defined over Fq. Then, F

is absolutely or geometrically irreducible if F(X ,Y,Z) is irreducible in Fq[X ,Y,Z].

58 Chapter 2. Projective algebraic curves defined over finite fields

2.2 Fq-rational branches of plane curvesDefinition 2.2.1. The conjugate map of Fq((T )) is the automorphism

κ : Fq((T )) → Fq((T ))

∑αiT i ↦→ ∑αqi T i

of Fq((T )).

Remark 2.2.2. The following occurs.

1. κ is not an Fq-automorphism of Fq((T )).

2. If σ is an Fq-automorphism of Fq((T )), then κ ∘σ ∘κ−1 is also an Fq-automorphism ofFq((T )). In particular, in terms of Definition 1.1.17, if the elements (ζ1(T ), η1(T )) and(ζ2(T ), η2(T )) of Fq[[T ]]2 are equivalent (resp. the element (ζ (T ), η(T )) of Fq[[T ]]2 isprimitive), then (κ(ζ1(T )),κ(η1(T ))) and (κ(ζ2(T )),κ(η2(T ))) are also equivalent (resp.(κ(ζ (T )),κ(η(T ))) is also primitive).

Definition 2.2.3. The Frobenius image Φ(b) of a branch b is the branch with primitive repre-sentation

Φ(ă) := (κ(α(T )) : κ(β (T )) : κ(γ(T ))),

where ă = (α(T ) : β (T ) : γ(T )) is a primitive representation of b.

Definition 2.2.4. A branch b is Fq-rational if it has a primitive representation ă with specialcoordinates (α(T ) : β (T ) : γ(T )) satisfying κ(α(T )) = α(T ), κ(β (T )) = β (T ) and κ(γ(T )) =

γ(T ). In particular, Φ(b) = b.

Remark 2.2.5. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Comments on Page 279).Let b be an Fq-rational branch with primitive representation ă = (α(T ) : β (T ) : γ(T )) in specialcoordinates satisfying κ(α(T )) = α(T ), κ(β (T )) = β (T ) and κ(γ(T )) = γ(T ). Then, anotherprimitive branch representation ă1 = (α1(T ) : β1(T ) : γ1(T )) in special coordinates of b does notsatisfy the property κ(α1(T )) = α1(T ), κ(β1(T )) = β1(T ) and κ(γ1(T )) = γ1(T ) unless thereexists an Fq-automorphism σ of Fq[[T ]] such that σ(Fq[[T ]])⊆ Fq[[T ]].

Theorem 2.2.6. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 8.9). Let F bean absolutely irreducible plane curve defined over Fq and consider P a nonsingular point of F .Then P ∈ P2(Fq) if and only if the unique branch of F centered at P is Fq-rational.

Theorem 2.2.7. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 8.10). Let F bean absolutely irreducible plane curve defined over Fq and consider P ∈ P2(Fq) a singular pointof F . For a linear branch b of F centered at P, b is Fq-rational if and only if the tangent line ofb is defined over Fq.

Theorem 2.2.8. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 8.13). If b is abranch of an absolutely irreducible plane curve F defined over Fq, then b is Fqn-rational forsome positive integer n.

2.3. Fq-rational function fields and Fq-rational places 59

2.3 Fq-rational function fields and Fq-rational placesLet F : F(X ,Y ) = 0 be an absolutely irreducible plane curve defined over Fq and, in

terms of Remark 1.1.43, let Fq(F ) = Fq(x,y) be the function field of F , where (x : y : 1) is ageneric point of F .

Definition 2.3.1. The Fq-rational function field of F , denoted by Fq(F ), is the subfield ofFq(F ) described by{

A(x,y)B(x,y)

: A(X ,Y ),B(X ,Y ) ∈ Fq[X ,Y ] and B(x,y) = 0}.

The elements of Fq(F ) are called Fq-rational functions of Fq(F ).

Definition 2.3.2. IfC(1)(X ,Y ) := ∑α

qikX iY k

for each C(X ,Y ) = ∑αikX iY k ∈ Fq[X ,Y ], then the conjugate map of Fq(F ) is the automorphism

ι : Fq(F ) → Fq(F )

f = A(x,y)B(x,y) ↦→ f (1) := A(1)(x,y)

B(1)(x,y)

of Fq(F ).

Remark 2.3.3. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Lemmas 8.14 and 8.15).1. The definition of ι does not depend on the representation of the functions f ∈ Fq(F ).

2. ι( f ) = f if and only if f ∈ Fq(F ).

Definition 2.3.4. Let P be a place of Fq(F ) and let b be the corresponding branch of F . TheFrobenius image of P , denoted by Φ(P), is the place of Fq(F ) corresponding to the branchΦ(b).

Remark 2.3.5. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Comments on Page 283).Let τ be a primitive representation of a place P of Fq(F ). Then,

κ ∘ τ ∘ ι−1

is a primitive representation of Φ(P).

Definition 2.3.6. Let P be a place of Fq(F ). Then, P is an Fq-rational place if Φ(P) = P .

Corollary 2.3.7. By Theorem 2.2.8, if P is a place of Fq(F ), then P is Fqn-rational for somepositive integer n.

Theorem 2.3.8. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Lemma 8.17 and Theorem8.31). The place P of Fq(F ) is Fq-rational if and only if the corresponding branch b of F isFq-rational.


Theorem 2.3.9. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 8.18). Let P bea nonsingular point of F . Then P ∈ P2(Fq) if and only if the place P of Fq(F ) correspondingto the unique branch b of F centered at P is Fq-rational.

2.3.1 Some remarks

In terms of the notation and the correspondence briefly introduced in Subsection 1.2.3,let us consider the function field Fq(x,y)/Fq as a constant field extension of Fq(x,y)/Fq. Then,the following definition, which is equivalent to Definition 2.3.6, holds.

Definition 2.3.10. (GOLDSCHMIDT, 2003, Page 75). A place P of Fq(x,y)/Fq is an Fq-

rational place if P ∩Fq(x,y) is a place of degree one of Fq(x,y)/Fq, or, equivalently, if P =

ConFq/Fq(x,y)(P′) for some place of degree one P ′ of Fq(x,y)/Fq, where ConFq/Fq(x,y) is the

conorm map.

For more details on the topics and definitions above mentioned, we refer to (GOLD-SCHMIDT, 2003, Section 3.2) and (STICHTENOTH, 2009, Section 3.6) for instance.

2.4 Fq-rational divisors and linear seriesLet F : F(X ,Y ) = 0 be an absolutely irreducible plane curve defined over Fq and, in

terms of Remark 1.1.43, let Fq(F ) = Fq(x,y) be the function field of F , where (x : y : 1) is ageneric point of F .

Definition 2.4.1. For D = ∑P is a place of Fq(F )

nPP ∈ Div(Fq(F )), define

Φ(D) := ∑P is a place of Fq(F )

nPΦ(P).

Definition 2.4.2. Let D ∈ Div(Fq(F )). D is an Fq-rational divisor if Φ(D) = D. Further, thesubgroup of Div(Fq(F )) consisting of all Fq-rational divisors is denoted by Div(Fq(F )).

Remark 2.4.3. Let D ∈ Div(Fq(F )). If P is Fq-rational, for all P ∈ Supp(D), then D is anFq-rational divisor. However, the converse is not true in general. In fact, based on Corollary2.3.7, let P be a place of Fq(F ) and consider n the smallest positive integer such that P isFqn-rational. Then, the divisor

D = P +Φ(P)+ · · ·+Φn−1(P)

is Fq-rational even if P is not, that is, if n > 1. The divisor D is called the closed place of P .

Definition 2.4.4. For a place P of Fq(F ), consider the notation given as in Remark 2.4.3. Then,the degree of P is defined by

deg(P) := n.

2.5. Space curves defined over Fq 61

Proposition 2.4.5. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Lemmas 8.23 and8.24). If f ∈ Fq(F )∖{0}, then both div( f ) and div(d f ) are Fq-rational divisors. Conversely, iff ∈ Fq(F )∖{0} and div( f ) is an Fq-rational divisor, then f ∈ Fq(F ) up to a constant factor.

Definition 2.4.6. A linear series D on Fq(F ) is Fq-rational if D ∼= P(S )⊆ |E|, where S ⊆L (E) has a basis consisting of Fq-rational functions and E is an Fq-rational divisor.

Theorem 2.4.7. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 8.26). If E is anFq-rational divisor, then |E| is an Fq-rational linear series.

Corollary 2.4.8. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Remark 8.27). The canon-ical linear series is an Fq-rational linear series.

2.5 Space curves defined over Fq

Let F : F(X ,Y ) = 0 be an absolutely irreducible plane curve defined over Fq and letFq(F ) = Fq(x,y) be the function field of F , where (x : y : 1) is a generic point of F . Further,consider X an irreducible curve in PM(Fq) such that Fq(X ) = Fq(F ).

In terms of Notation 1.3.9, it is possible to establish the following definition.

Definition 2.5.1. A curve X is defined over Fq if it is given by a point

(x0 : · · · : xM) ∈ PM(Fq(F )),

with xi ∈ Fq(F ) for all i = 0, . . . ,M.

Notation 2.5.2. If X is defined over Fq, then X is called a geometrically irreducible curve

defined over Fq.

Definition 2.5.3. Suppose that X is defined over Fq and given by the point (x0 : · · · : xM) ∈PM(Fq(F )), with xi ∈ Fq(F ) for all i = 0, . . . ,M. A branch b of X associated to the place P

of Fq(F ) is Fq-rational if there exists a primitive representation τ of P such that the primitiverepresentation ă = (τ(x0) : · · · : τ(xM)) of b satisfies κ(τ(xi)) = τ(xi) for all i = 0, . . . ,M, whereκ is given as in Definition 2.2.1.

Theorem 2.5.4. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 8.29). Supposethat X is defined over Fq and that Q is a nonsingular point of X . Then, the following assertionsare equivalent.

1. Q ∈ PM(Fq).

2. The unique branch b of X centered at Q is Fq-rational.

3. The place P of Fq(F ) associated to b is Fq-rational.

Theorem 2.5.5. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 8.31). If X isdefined over Fq, then a branch b of X is Fq-rational if and only if the associated place P ofFq(F ) is Fq-rational.


2.5.1 Rational transformations and morphisms defined over Fq

Let F : F(X ,Y ) = 0 be an absolutely irreducible plane curve defined over Fq and letFq(F ) = Fq(x,y) be the function field of F , where (x : y : 1) is a generic point of F . Also,consider X a geometrically irreducible curve defined over Fq such that Fq(X ) = Fq(F ).

Definition 2.5.6. Let φ = ( f0 : · · · : fN) ∈ PN(Fq(F )) be a rational transformation on X . φ is

said to be defined over Fq if ( f0 : · · · : fN) ∈ PN(Fq(F )). If Y is the image curve of φ and φ

is defined over Fq, then Y is also defined over Fq and φ : X 99K Y is called an Fq-rational

covering.

Remark 2.5.7. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Remark 8.19). Let Y be ageometrically irreducible curve defined over Fq. If X and Y are birationally equivalent over Fq,then, in general, they do not have to be birationally equivalent over Fq as well.

2.5.2 Nonsingular models defined over Fq

Let F : F(X ,Y ) = 0 be an absolutely irreducible plane curve defined over Fq and letFq(F ) = Fq(x,y) be the function field of F , where (x : y : 1) is a generic point of F . Also,consider X a geometrically irreducible curve defined over Fq such that Fq(X ) = Fq(F ).

The following remark discuss the existence of nonsingular models of X defined overFq.

Remark 2.5.8. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Remark 8.30). Theorem2.4.7 provides a tool to construct nonsingular models defined over Fq of X . Particularly, if X

is neither rational, nor elliptic, nor hyperelliptic, the canonical curve is a nonsingular modeldefined over Fq of X .

Lastly, in addition to Proposition 1.3.34 and Notation 1.3.35, the following correspon-dences are a consequence of Theorem 2.5.5.

Proposition 2.5.9. Suppose that X is nonsingular. Then, there exist the following one-to-onecorrespondences:

Fq-rational places of Fq(X )1−1←→ Fq-rational branches of X

1−1←→ Fq-rational points on X .

2.6 The Zeta function of a curve defined over Fq

Let X be a nonsingular geometrically irreducible curve of genus g in PM(Fq) definedover Fq.

2.6. The Zeta function of a curve defined over Fq 63

Definition 2.6.1. Let D = ∑Q∈X

nQQ be a divisor of X . Then, it is possible to define

deg(D) := ∑Q∈X

nQdeg(Q),

where, for each Q ∈X , deg(Q) is considered as in Definition 2.4.4. Also, one may define

N (D) := qdeg(D) = ∏Q∈X

N (Q)nQ,

where N (Q) := qdeg(Q).

Definition 2.6.2. For a complex variable s, the Zeta function of X over Fq is defined by

ζ (X ,s) := ∑D is an effective Fq-rational divisor

N (D)−s.

Theorem 2.6.3 (Euler product). For s ∈ C such that Re(s)> 1, the product

∏Q∈X

(1−N (Q)−s)−1 (2.1)

is absolutely convergent to the function ζ (X ,s). In particular, the product (2.1) is independentof the order of its factors.


Theorem 2.6.4. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 9.5). For s ∈ Csuch that Re(s)> 1,

ζ (X ,s) = L(q−s)

(1−q−s)(1−q1−s),

where L(T ) =2g

∑i=0

aiT i ∈ Z[T ], with a0 = 1 and a2g = qg.

Theorem 2.6.5 (Functional equation). For s ∈ C such that Re(s)> 1,

q(g−1)(2s−1)ζ (X ,s) = ζ (X ,1− s).


Definition 2.6.6. For s ∈ C such that Re(s)> 1, define

t = t(s) := q−s.

Definition 2.6.7. For s ∈ C such that Re(s)> 1, consider t given as in Definition 2.6.6. Then,one can denote by

Z(X , t)

the expression of ζ (X ,s), where q−s is replaced by t.


Theorem 2.6.8. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 9.7). Consider tgiven as in Definition 2.6.6. If |t|< q−1, then

Z(X , t) = exp(

∞

∑n=1

Nqn(X )

ntn),

where Nqn(X ) denotes the number of Fqn-rational points on X for each positive integer n.

Proposition 2.6.9. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Remarks on Page 339).Consider t given as in Definition 2.6.6. If |t|< q−1, then

Z(X , t) = 1+∞

∑n=1

Antn,

where An is the number of effective Fq-rational divisors of X of degree n.

The following result is a straightforward consequence of Theorem 2.6.4 and Definition2.6.6.

Proposition 2.6.10. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Comments on Page339). Consider t given as in Definition 2.6.6. If |t|< q−1, then

Z(X , t) = L(t)(1− t)(1−qt)

where

L(T ) = Lq(T ) =

1, if g = 0

1+2g−1

∑i=1

aiT i +a2gT , otherwise(2.2)

belongs to Z[T ].

Definition 2.6.11. The polynomial given in (2.2) is the L-polynomial of X over Fq.

Proposition 2.6.12. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Proposition 9.9). TheL-polynomial of X over Fq given by (2.2) satisfies the following properties:

1. L(T ) = qgT 2gL((qT )−1).

2. a2g−i = qg−iai for all i = 0, . . . ,g.

Theorem 2.6.13. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 9.10). Considerthe factorization

L(T ) =2g

∏i=1

(1−ωiT )

of the L-polynomial of X over Fq given in (2.2). Then,

Nqn(X ) = qn +1−2g

∑i=1

ωni ,

where Nqn(X ) is the number of Fqn-rational points on X for each positive integer n.

2.7. The Hasse-Weil theorem and maximal curves 65

Proposition 2.6.14. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Proposition 9.12). Foreach i ∈ {1, . . . ,g}, the coefficient ai of the L-polynomial of X over Fq given in (2.2) satisfiesthe following relation:

iai = Ni +Ni−1a1 + · · ·+N1ai−1,

where Ni = Nqi(X )−(qi+1) and, for each i∈ {1, . . . ,g}, Nqi(X ) is the number of Fqi-rationalpoints on X .

Proposition 2.6.15. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Propositions 9.15 and9.16). Considering the factorization

L(T ) =2g

∏i=1

(1−ωiT )

of the L-polynomial of X over Fq given in (2.2), the L-polynomial Lqn(T ) of X over Fqn isgiven by

Lqn(T ) =2g

∏i=1

(1−ωni T ).

Now, this section ends with the following important result.

Theorem 2.6.16 (Serre). Let Y be a nonsingular geometrically irreducible curve defined overFq. Suppose that there exists an Fq-rational covering φ : X → Y . Then, the L-polynomial of Y

over Fq divides the L-polynomial of X over Fq.

Proof. A proof of this result can be found in (AUBRY; PERRET, 2004). �

2.7 The Hasse-Weil theorem and maximal curves

Let X be a nonsingular geometrically irreducible curve of genus g in PM(Fq) definedover Fq. Further, let us consider the factorization

L(T ) =2g

∏i=1

(1−ωiT )

of the L-polynomial of X over Fq.

The following theorem, which is the Riemann hypothesis for irreducible curves definedover finite fields, is fundamental in the theory. It was first proved by Hasse for g = 1 (HASSE,1936a), (HASSE, 1936b), (HASSE, 1936c) and then by Weil for g > 1 (WEIL, 1948).

Theorem 2.7.1 (Hasse-Weil). For each i ∈ {1, . . . ,2g},

|ωi|= q1/2.


Proof. A proof of this result can be found in (HIRSCHFELD; KORCHMÁROS; TORRES,2008, Section 9.2). �

As a consequence of the Hasse-Weil theorem (Theorem 2.7.1), the following bound forthe number Nq(X ) of Fq-rational points on X can be obtained from Theorem 2.6.13:

Theorem 2.7.2 (Hasse-Weil bound).

|Nq(X )− (q+1)|6 2gq1/2. (2.3)

Proof. A proof of this result can be found in (HIRSCHFELD; KORCHMÁROS; TORRES,2008, Section 9.2). �

The previous theorem leads us to the following definition.

Definition 2.7.3. The curve X is called maximal (resp. minimal) over Fq if it attains the Hasse-Weil upper (resp. lower) bound (2.3), or, equivalently, if ωi =−q1/2 (resp. ωi = q1/2) for eachi ∈ {1, . . . ,2g}.

Finally, the following result is an immediate consequence of Theorem 2.6.16 and Defini-tion 2.7.3.

Theorem 2.7.4 (Serre). Let Y be a nonsingular geometrically irreducible curve defined over Fq.Suppose that there exists an Fq-rational covering φ : X → Y . If X is either Fq2n-maximal orFq2n-minimal, then X is Fq2n-maximal (resp. Fq2n-minimal) if and only if Y is Fq2n-maximal(resp. Fq2n-minimal).

67

CHAPTER

3THE STÖHR-VOLOCH THEORY

Based on (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Chapters 7 and 8), (TOR-RES, 2000, Sections 1 and 2) and (STÖHR; VOLOCH, 1986), the purpose of this chapter is toestablish a general background on the Stöhr-Voloch Theory.

Considering p a prime number and q = pm for some positive integer m, the followingtheorem, whose proof follows essentially by an application of Bézout’s Theorem (see Theorem1.1.36), is a slightly simpler version of the main result of (STÖHR; VOLOCH, 1986). Nowadays,this is sometimes called the Baby Stöhr-Voloch Theorem.

Theorem 3.0.1. (STÖHR; VOLOCH, 1986, Theorem 0.1). Let p > 2 and let F : F(X ,Y,Z) = 0be an absolutely irreducible plane curve of degree d defined over Fq. If F has a finite number ofinflection points and Nq(F ) is defined by

Nq(F ) := #{(α : β : γ) ∈ P2(Fq) : F(α,β ,γ) = 0

},

then Nq(F ) satisfies

Nq(F )6d(d +q−1)

2.

3.1 Morphisms and Weierstrass points

Throughout this section, let K be an algebraically closed field of characteristic p> 0 andlet X be a nonsingular irreducible curve of genus g over K. Also, let

φ = ( f0 : · · · : fN) : X → PN(K)

68 Chapter 3. The Stöhr-Voloch theory

be a nondegenerate morphism on X and, as discussed on Subsection 1.3.3.2, let D be thecorresponding base-point-free linear series of dimension N and degree d, that is, let

D =

{div( N

∑i=0

αi fi

)+E : (α0 : · · · : αN) ∈ PN(K)

}=

{φ*(∆) : ∆ is a hyperplane in PN(K)

},

where E = ∑Q∈X

eQQ is defined by

eQ =−min{

vQ( f0), . . . ,vQ( fN)

}

and φ*(∆) = div

( N

∑i=0

αi fi

)+E is the pullback of the hyperplane ∆ :

N

∑i=0

αiXi = 0 by φ .

3.1.1 Hermitian invariants and osculating spaces

Definition 3.1.1. Let Q ∈X . An integer j is a Hermitian Q-invariant or a (D ,Q)-order if thereexists a divisor D ∈D such that vQ(D) = j.

From Proposition 1.3.37, the following occurs.

Proposition 3.1.2. Let Q ∈X . An integer j is a (D ,Q)-order if and only if D j(Q)) D j+1(Q).In particular, there are exactly N +1 (D ,Q)-orders, namely 0 = j0 < · · ·< jN 6 d.

Definition 3.1.3. Let Q ∈X and let 0 = j0 < · · · < jN 6 d be given as in Proposition 3.1.2.Then ( j0, . . . , jN) is the sequence of (D ,Q)-orders.

As a consequence of the Riemann-Roch Theorem (see Theorem 1.2.61), the followingresult holds.

Proposition 3.1.4. Suppose that D is the canonical linear series. For each Q ∈ X , j is a(D ,Q)-order if and only if j+1 is a gap number of Q.

Definition 3.1.5. Let Q∈X . For each i∈ {0, . . . ,N−1}, the i-th osculating space at Q, denotedby Li(Q), is defined by

Li(Q) :=⋂

∆ is a hyperplane in PN (K)

such that vQ(φ*(∆))> ji+1

∆,

where ( j0, . . . , jN) is the sequence of (D ,Q)-orders. In the particular case i = N−1, LN−1(Q) iscalled the osculating hyperplane at Q.

It follows from the definitions that, for each Q ∈X , L0(Q)⊆ ·· · ⊆ LN−1(Q). Further-more, the following occurs.

3.1. Morphisms and Weierstrass points 69

Theorem 3.1.6. (STÖHR; VOLOCH, 1986, Theorem 1.1 and Scholium 1.2). Let Q ∈X , let tbe a local parameter at Q, and let ( j0, . . . , jN) be the sequence of (D ,Q)-orders, where j0 = 0.

1. For each i> 1,

ji(Q) = min{

j : (D( j0)t t

eQ f0(Q),...,D( j0)t t

eQ fN(Q)),...,(D( ji−1)t t

eQ f0(Q),...,D( ji−1)t t

eQ fN(Q)),

(D( j)t t

eQ f0(Q),...,D( j)t t

eQ fN(Q)) are linearly independent over K

}.

2. Let Si(Q) be the K-linear subspace of KN+1 generated by

(D( j0)t teQ f0(Q), . . . ,D( j0)

t teQ fN(Q)), . . . ,(D( ji)t teQ f0(Q), . . . ,D( ji)

t teQ fN(Q)).

Then, Li(Q) = P(Si(Q))⊆ PN(K). In particular, dim(Li(Q)) = i.

3. Let k0 < · · ·< ki be non-negative integers, with i6 N, such that the vectors

(D(k0)t teQ f0(Q), . . . ,D(k0)

t teQ fN(Q)), . . . ,(D(ki)t teQ f0(Q), . . . ,D(ki)

t teQ fN(Q))

are linearly independent over K. Then jl 6 kl , for each l ∈ {0, . . . , i}.

Corollary 3.1.7. (STÖHR; VOLOCH, 1986, Corollary 1.3). Let Q ∈X , let t be a local pa-rameter at Q, and let ( j0, . . . , jN) be the sequence of (D ,Q)-orders, where j0 = 0. Then, theosculating hyperplane at Q is given by the equation

det

X0 · · · XN

D( j0)t teQ f0(Q) · · · D( j0)

t teQ fN(Q)...

...

D( jN−1)t teQ f0(Q) · · · D( jN−1)

t teQ fN(Q)

= 0.

3.1.2 Order sequence and ramification divisor

Now, the study of the linear series D at a general point is addressed. For this, let Q0 ∈X

be a fixed element such that eQ0 = 0, and let t0 be a local parameter at Q0. Note that, in particular,t0 is a separating variable of K(X ) (see Proposition 1.2.24).

Let M > 0 be any non-negative integer.

Definition 3.1.8. A wronskian on X is a function of K(X ) of type

W k0,...,kMg0,...,gM ;y := det

D(k0)

y g0 · · · D(k0)y gM

......

D(kM)y g0 · · · D(kM)

y gM

,where (k0, . . . ,kM) is a sequence of non-negative integers satisfying k0 < · · ·< kM, g0, . . . ,gM ∈K(X ), and y ∈K(X ) is a separating variable.


Definition 3.1.9. If g0, . . . ,gM ∈K(X ) and y is a separating variable of K(X ), then one maydefine

A (g0, . . . ,gM ; y) :={(k0, . . . ,kM) ∈ NM+1 : k0 < · · ·< kM and W k0,...,kM

g0,...,gM ;y = 0}.

Definition 3.1.10. In NM+1, the lexicographic order is the partial ordering <lex

given as follows.

Let (k0, . . . ,kM) and (l0, . . . , lM)∈NM+1. Then, (k0, . . . ,kM) <lex

(l0, . . . , lM) if in (l0−k0, . . . , lM−

kM) ∈ ZM+1 the leftmost nonzero entry is positive.

Proposition 3.1.11. (COX; LITTLE; O’SHEA, 2007, Proposition 4 on Chapter 2). The lexico-graphic order <

lexin NM+1 is a well-ordering. In particular, every non-empty set in NM+1 has a

minimum element with respect to <lex

.

Now, consider the wronskians of the form

W k0,...,kNf0,..., fN ; t0

.

Proposition 3.1.12. Let ( j0, . . . , jN) be the sequence of (D ,Q0)-orders, where j0 = 0. Then( j0, . . . , jN) ∈A ( f0, . . . , fN ; t0). In particular, A ( f0, . . . , fN ; t0) has a minimum element withrespect to the lexicographic order, namely (ε0, . . . ,εN), where ε0 = 0.

Proof. The result follows from Theorem 3.1.6 and Proposition 3.1.11. �

Proposition 3.1.13. (STÖHR; VOLOCH, 1986, Considerations on Page 5). Let (ε0, . . . ,εN) begiven as in Proposition 3.1.12. Similarly to items 1 and 3 of Theorem 3.1.6, the following occurs.

1. For each i> 1,

εi = min{

ε : (D(ε0)t0

f0,...,D(ε0)t0

fN),...,(D(εi−1)t0

f0,...,D(εi−1)t0

fN),

(D(ε)t0

f0,...,D(ε)t0

fN) are linearly independent over K(X )

}.

2. Let k0 < · · ·< ki be non-negative integers, with i6 N, such that the vectors

(D(k0)t0

f0, . . . ,D(k0)t0

fN), . . . ,(D(ki)t0

f0, . . . ,D(ki)t0

fN)

are linearly independent over K(X ). Then εl 6 kl , for each l ∈ {0, . . . , i}.

Proposition 3.1.14. (STÖHR; VOLOCH, 1986, Proposition 1.4). Let (ε0, . . . ,εN) be given asin Proposition 3.1.12.

1. For each i ∈ {0, . . . ,N}, let f ′i =N

∑k=0

αik fk, where (αik) ∈ GLN+1(K). Then,

W ε0,...,εNf ′0,..., f

′N ; t0

= det(αik)Wε0,...,εNf0,..., fN ; t0

.

2. If f ∈K(X ), then

W ε0,...,εNf f0,..., f fN ; t0

= f N+1W ε0,...,εNf0,..., fN ; t0

.

3.1. Morphisms and Weierstrass points 71

3. If x ∈K(X ) is another separating variable, then

W ε0,...,εNf0,..., fN ;x =

(dt0dx

)ε0+···+εN

W ε0,...,εNf0,..., fN ; t0

.

Remark 3.1.15. Proposition 3.1.14 shows that the sequence (ε0, . . . ,εN) given in Proposition3.1.12 depends only on the linear series D . In other words, for each (αik) ∈ GLN+1(K), f ∈

K(X )∖{0} and x ∈K(X ) a separating variable, if f ′i =N

∑k=0

αik f fk, with i ∈ {0, . . . ,N}, then

A ( f ′0, . . . , f ′N ; x) = /0,

and the minimum with respect to the lexicographic order in A ( f ′0, . . . , f ′N ; x) is given by(ε0, . . . ,εN).

Definition 3.1.16. Let (ε0, . . . ,εN) be given as in Proposition 3.1.12 and Remark 3.1.15. Foreach i ∈ {0, . . . ,N}, εi is a D-order, and the sequence (ε0, . . . ,εN) is the sequence of D-orders.Further, D is called classical if (ε0, . . . ,εN) = (0, . . . ,N). Otherwise, D is called nonclassical.

From now until the end of this subsection, let us consider the wronskian

W ε0,...,εNf0,..., fN ;x,

where x is a separating variable of K(X ). Using the local expansion of a function at a localparameter, one can easily obtain the following result.

Lemma 3.1.17. (STÖHR; VOLOCH, 1986, Proof of Theorem 1.5). Let Q be a point of X suchthat x is a local parameter at Q ∈X and eQ = 0. If ( j0, . . . , jN) is the sequence of (D ,Q)-orders,then

vQ(Wε0,...,εNf0,..., fN ;x)>

N

∑i=0

( ji− εi),

with the equality holding if and only if

det((

jik

))≡ 0 (mod p).

If Q ∈X is an arbitrary point, let t be a local parameter at Q. Applying Proposition3.1.14, one may obtain that

W ε0,...,εNf0,..., fN ;x =

(dtdx

)ε0+···+εN

W ε0,...,εNf0,..., fN ; t

=

(dtdx

)ε0+···+εN(

1teQ

)N+1

W ε0,...,εNteQ f0,...,t

eQ fN ; t


and thus

vQ(Wε0,...,εNf0,..., fN ;x) =−(ε0 + · · ·+ εN)vQ(div(dx))− (N +1)vQ(E)+ vQ(W

ε0,...,εNteQ f0,...,t

eQ fN ; t),

where

−min{

vQ(teQ f0), . . . ,vQ(t

eQ fN)

}= 0.

Now, it is possible to define the divisor

DRam := div(W ε0,...,εNf0,..., fN ;x)+(ε0 + · · ·+ εN)div(dx)+(N +1)E.

As a consequence of Proposition 3.1.14 and the previous considerations, the followingoccurs.

Proposition 3.1.18. The divisor DRam depends only on the linear series D . Further, the multi-plicity of DRam at the point Q ∈X is given by the multiplicity of the wronskian

W ε0,...,εNteQ f0,...,t

eQ fN ; t

at Q, where t is a local parameter at Q.

Definition 3.1.19. The divisor DRam is the ramification divisor of D .

From Lemma 3.1.17 and Proposition 3.1.18, the ramification divisor of D satisfies thefollowing property.

Theorem 3.1.20. (STÖHR; VOLOCH, 1986, Theorem 1.5). If ( j0, . . . , jN) is the sequence of(D ,Q)-orders, then

vQ(DRam)>N

∑i=0

( ji− εi),


det((

jik

))≡ 0 (mod p).

Corollary 3.1.21. The ramification divisor DRam has the following properties:

1. DRam is an effective divisor.

2. For each Q∈X , vQ(DRam) = 0 if and only if ji = εi for all i= 0, . . . ,N, where ( j0, . . . , jN)

is the sequence of (D ,Q)-orders. In particular, (ε0, . . . ,εN) is the sequence of (D ,Q)-orders for almost all points Q ∈X .

Definition 3.1.22. A point Q ∈X is a D-ordinary point if its sequence of (D ,Q)-orders isequal to (ε0, . . . ,εN). Otherwise, Q ∈X is a D-Weierstrass point. In particular,

Supp(DRam) =

{Q ∈X : Q is a D-Weierstrass point

}.

3.2. The Stöhr-Voloch theorem 73

This section ends with the following criterion for the classicality of D .

Proposition 3.1.23. (STÖHR; VOLOCH, 1986, Corollaries 1.7 and 1.8). Let Q ∈X and let( j0, . . . , jN) be the sequence of (D ,Q)-orders. If the integer

∏i>k

ji− jki− k

is not divisible by p, then D is classical and

vQ(DRam) =N

∑i=0

( ji− i).

In particular, if p > d or p = 0, then D is classical.

3.2 The Stöhr-Voloch theoremThroughout this section, let p> 0 be a prime number, let Fq be the finite field with q= pm

elements and let Fq be the algebraic closure of Fq. Also, let X be a nonsingular irreduciblecurve of genus g defined over Fq, let

φ = ( f0 : · · · : fN) : X → PN(Fq)

be a nondegenerate morphism on X , with f0, . . . , fN ∈ Fq(X ), and let D be the correspondingbase-point-free linear series of dimension N and degree d, that is, let

D =

{div( N

∑i=0

αi fi

)+E : (α0 : · · · : αN) ∈ PN(Fq)

}=

{φ*(∆) : ∆ is a hyperplane in PN(Fq)

},

where E = ∑Q∈X

eQQ is defined by

eQ :=−min{

vQ( f0), . . . ,vQ( fN)

}

and φ*(∆) = div

( N

∑i=0

αi fi

)+E is the pullback of the hyperplane ∆ :

N

∑i=0

αiXi = 0 by φ . Note

here that E and D are defined over Fq.

In order to obtain an upper bound for the number of Fq-rational points on X , one mayconsider the possibly larger set of all points Q ∈X such that the image of φ(Q) under theFrobenius map of PN(Fq) is in the osculating hyperplane LN−1(Q) at Q. From Corollary 3.1.7, apoint Q ∈X with eQ = 0 is in this set if and only if

det

f0(Q)q · · · fN(Q)q

D( j0)t f0(Q) · · · D( j0)

t fN(Q)...

...

D( jN−1)t f0(Q) · · · D( jN−1)

t fN(Q)

= 0,


where t is a local parameter at Q.

Thus, if x ∈ Fq(X ) is a separating variable, then one is lead to study determinants of theform

det

f q0 · · · f q

N

D(k0)x f0 · · · D(k0)

x fN...

...

D(kN−1)x f0 · · · D(kN−1)

x fN

,where k0 < · · ·< kN−1 are non-negative integers.

Thus, let M > 0 be a non-negative integer.

Definition 3.2.1. If (k0, . . . ,kM−1) is a sequence of non-negative integers satisfying k0 < · · ·<kM−1, g0, . . . ,gM ∈ Fq(X ), and y ∈ Fq(X ) is a separating variable, define

W k0,...,kM−1y (g0, . . . ,gM) := det

gq

0 · · · gqM

D(k0)y g0 · · · D(k0)

y gM...

...

D(kM−1)y g0 · · · D(kM−1)

y gM

.

Definition 3.2.2. If g0, . . . ,gM ∈ Fq(X ) and y is a separating variable of Fq(X ), then one maydefine

B(g0, . . . ,gM ; y) :={(k0, . . . ,kM−1) ∈ NM : k0 < · · ·< kM−1 and W k0,...,kM−1

y (g0, . . . ,gM) = 0}.

Proposition 3.2.3. (STÖHR; VOLOCH, 1986, Proposition 2.1). Considering the previousnotation, B( f0, . . . , fN ; x) = /0. Therefore, in terms of Proposition 3.1.11, let (ν0, . . . ,νN−1) bethe minimum of B( f0, . . . , fN ; x) with respect to the lexicographic order. Then there exists aninteger i ∈ {1, . . . ,N} such that

νk =

{εk, if k < i

εk+1, otherwise,

where (ε0, . . . .εN) is the sequence of D-orders.

Proposition 3.2.4. (STÖHR; VOLOCH, 1986, Considerations on Page 9). Let (ν0, . . . ,νN−1)

be given as in Proposition 3.2.3 and let k0 < · · ·< ki be non-negative integers, with i6 N−1,such that the vectors

( f q0 , . . . , f q

N),(D(k0)x f0, . . . ,D

(k0)x fN), . . . ,(D

(ki)x f0, . . . ,D

(ki)x fN)

are linearly independent over Fq(X ). Then νl 6 kl , for each l ∈ {0, . . . , i}.

Similarly to Proposition 3.1.14, the following holds.


Proposition 3.2.5. (STÖHR; VOLOCH, 1986, Proposition 2.2). Let (ν0, . . . ,νN−1) be given asin Proposition 3.2.3.

1. For each i ∈ {0, . . . ,N}, let f ′i =N

∑k=0

αik fk, where (αik) ∈ GLN+1(Fq). Then,

W ν0,...,νN−1x ( f ′0, . . . , f ′N) = det(αik)W

ν0,...,νN−1x ( f0, . . . , fN).

2. If f ∈ Fq(X ), then

W ν0,...,νN−1x ( f f0, . . . , f fN) = f N+qW ν0,...,νN−1

x ( f0, . . . , fN).

3. If x′ ∈ Fq(X ) is another separating variable, then

W ν0,...,νN−1x′ ( f0, . . . , fN) =

(dxdx′

)ν0+···+νN−1

W ν0,...,νN−1x ( f0, . . . , fN).

Corollary 3.2.6. The sequence (ν0, . . . ,νN−1) given in Proposition 3.2.3 depends only on thelinear series D and on q.

Therefore, it is possible to establish the following definition.

Definition 3.2.7. Let (ν0, . . . ,νN−1) be given as in Proposition 3.2.3. For each i∈ {0, . . . ,N−1},νi is a Fq-Frobenius order of D , and the sequence (ν0, . . . ,νN−1) is the sequence of Fq-Frobenius

orders of D . Further, D is called Fq- Frobenius classical if (ν0, . . . ,νN−1) = (0, . . . ,N − 1).Otherwise, D is called Fq- Frobenius nonclassical.

Now, defining the divisor

DSV := div(W ν0,...,νN−1x ( f0, . . . , fN))+(ν0 + · · ·+νN−1)div(dx)+(N +q)E

of degree

(ν0 + · · ·+νN−1)(2g−2)+(N +q)d,

as a consequence of Proposition 3.2.5, the following result occurs.

Proposition 3.2.8. The divisor DSV depends only on the linear series D and on q. Further, themultiplicity of DSV at a point Q ∈X is given by the multiplicity of the function

W ν0,...,νN−1t (teQ f0, . . . , t

eQ fN)

at Q, where t ∈ Fq(X ) is a local parameter at Q. In particular, DSV is an effective divisor andthe following holds:

1. Let Q∈X be such that its sequence of (D ,Q)-orders ( j0, . . . , jN) satisfies ( j0, . . . , jN−1)=

(ν0, . . . ,νN−1). Then Q ∈ Supp(DSV) if and only if the image of φ(Q) under the Frobeniusmap of PN(Fq) is contained in the osculating hyperplane at Q.


2. Let Q ∈X be such that its sequence of (D ,Q)-orders ( j0, . . . , jN) satisfies νi < ji. ThenQ ∈ Supp(DSV).

3. Every Fq-rational point of X is in the support of DSV.

Definition 3.2.9. The divisor DSV is the Stöhr-Voloch divisor of D .

Using the local expansion of a function at a local parameter, one can obtain the followingresult.

Proposition 3.2.10. (STÖHR; VOLOCH, 1986, Proposition 2.4). For Q ∈X , let ( j0, . . . , jN)

be its sequence of (D ,Q)-orders.

1. If Q is a Fq-rational point of X , then

vQ(DSV)>N

∑i=1

( ji−νi−1),


det((

jiνk

))16i6N

06k6N−1

≡ 0 (mod p).

2. If Q is an arbitrary point of X , then

vQ(DSV)>N−1

∑i=1

( ji−νi),

with the strictly inequality holding if

det((

jiνk

))06i6N−106k6N−1

≡ 0 (mod p).

Proposition 3.2.11. (STÖHR; VOLOCH, 1986, Corollary 2.6). If Q is a Fq-rational point ofX and ( j0, . . . , jN) is its sequence of (D ,Q)-orders, then

vQ(DSV)> N j1.

As a consequence, one may obtain the following upper bound for the number Nq(X ) ofFq-rational points on X , which is a refinement of (STÖHR; VOLOCH, 1986, Theorem 2.13)obtained from some remarks at the beginning of (STÖHR; VOLOCH, 1986, Section 3).

Theorem 3.2.12 (Stöhr-Voloch).

Nq(X )6(ν0 + · · ·+νN−1)(2g−2)+(N +q)d−∑A(Q)

N,

where

A(Q) =

N

∑i=1

( ji(Q)−νi−1)−N, if Q is a Fq rational point on X

N−1

∑i=0

( ji(Q)−νi), otherwise.


Finally, this section ends with the following criterion for the Fq-Frobenius classicality ofD .

Proposition 3.2.13. (STÖHR; VOLOCH, 1986, Corollary 2.7). Let Q be an Fq-rational pointon X and let ( j0, . . . , jN) be the sequence of (D ,Q)-orders. If the integer

∏i>k

ji− jki− k

is not divisible by p, then D is Fq-Frobenius classical and

vQ(DSV) = N +N

∑i=1

( ji− i).

In particular, if p > d, then D is Fq-Frobenius classical.

Part II


81

CHAPTER

4PLANE SECTIONS OF FERMAT SURFACES

OVER FINITE FIELDS

The present chapter is an adapted version of (BORGES; COOK; COUTINHO, 2018).Here, in some excerpts, further details and explanations are provided.

4.1 IntroductionLet F be the plane curve obtained by the intersection of the Fermat surface

S : Xd0 +Xd

1 +Xd2 +Xd

3 = 0

with the planeP : X3− e0X0− e1X1− e2X2 = 0,

where d is a positive integer, e0,e1,e2 ∈ Fq, and Fq is the finite field with q = pm elements, for aprime number p. In other words, let

F : Xd0 +Xd

1 +Xd2 +(e0X0 + e1X1 + e2X2)

d = 0. (4.1)

Characterizing this general plane curve in terms of its rational points and its irreducibleand nonsingular components presents many challenges. For instance, the case p = 2 and e0 =

e1 = e2 = 1 has been extensively investigated over the past decades (see (van LINT; WILSON,1986), (JANWA; WILSON, 1993), (JANWA; MCGUIRE; WILSON, 1995) and (HERNANDO;MCGUIRE, 2011)), where in particular the following result was essential in Hernando andMcGuire’s proof of an important conjecture regarding exceptional numbers.

Theorem. (HERNANDO; MCGUIRE, 2011). The polynomial

Xd0 +Xd

1 +Xd2 +(X0 +X1 +X2)

d

(X0 +X1)(X0 +X2)(X1 +X2)

82 Chapter 4. Plane sections of Fermat surfaces over finite fields

has an absolutely irreducible factor defined over F2 for all d not of the form d = 2i + 1 ord = 22i−2i +1.

In this chapter, the problem of studying the plane curve given in (4.1) is considered froma somewhat different point of view. Based on techniques developed by Carlin and Voloch in(CARLIN; VOLOCH, 2004), the plane curve

C : C(X0,X1,X2) = Xd0 +Xd

1 +Xd2 +(e0X0 + e1X1 + e2X2)

d = 0, (4.2)

is characterized in cases where q = pm = 2d +1 is a prime power, p > 3, and e0, e1 and e2 arearbitrary elements in Fq. More precisely, a complete description of its irreducible and nonsingularcomponents is given, and also their number of Fq-rational points is provided. Consequently,a family of plane curves attaining the Stöhr-Voloch bound is constructed and the followingtheorem, which is the main result of this chapter, is proved.

Theorem 4.1.1. If C is not the union of d lines, then the following statements hold.

1. C is the union of n ∈ {0,1,2,3} lines defined over Fq and a nonsingular classical planecurve F of degree d.

2. The possibilities for the number of Fq-rational points on F are

12

d(d+q−1)− 12

i(d−2),

with i ∈ {0,1,2,3,d,3d}. In particular, F meets the Stöhr-Voloch bound in (4.7).

It is worth mentioning that few examples of curves attaining the Stöhr-Voloch bound areknown. Such explicit constructions are of interest in areas such as finite geometry and codingtheory.

This chapter is organized as follows. In Section 4.4, the Fq-points and linear componentsof C are detailed. In particular, it is shown that if C is not the union of d lines, then

C(X0,X1,X2) = (e0X0 + e1X1)i0(e0X0 + e2X2)

i1(e1X1 + e2X2)i2F(X0,X1,X2),

where i0, i1, i2 ∈ {0,1} and F(X0,X1,X2) has no linear factors. In Section 4.5, it is shown thatthe plane curve F : F(X0,X1,X2) = 0 is Fq-disjoint from any linear component of C . Thesefacts are used to prove Theorem 4.1.1, which relies on important results obtained in (STÖHR;VOLOCH, 1986), (VILLEGAS; VOLOCH, 1999) and (CARLIN; VOLOCH, 2004). Full detailsare given in Subsection 4.6.1.

4.2 NotationThe following notation is used throughout this chapter.

∙ Here, Fq ∖{0} is denoted by F*q.

4.3. Rudiments of the Stöhr-Voloch theory 83

∙ The set of points of P2(Fq) on a plane curve G defined over Fq is denoted by G (Fq) andits cardinality is denoted by #G (Fq).

∙ The quadratic character of Fq is denoted by η. That is,

η(α) =

1, if α is a nonzero square0, if α = 0−1, if α is a non-square,

for all α ∈ Fq. In particular, since d = q−12 , it follows that η(α) = αd , for all α ∈ Fq.

∙ The notation {η(e0),η(e1),η(e2)}= {0,1,1} means that two of the values η(e0), η(e1),η(e2) are equal to 1 and one is equal to 0; similarly for the other cases.

∙ The points (1 : 0 : 0), (0 : 1 : 0) and (0 : 0 : 1) in P2(Fq) are denoted by P0, P1 and P2,respectively. The points (−e1 : e0 : 0), (−e2 : 0 : e0) and (0 : −e2 : e1) are denoted byP01 = P10, P02 = P20 and P12 = P21, respectively.

∙ The sets {(0 : α1 : 1) ∈ P2(Fq) : η(α1) =−1

},

{(α0 : 0 : 1) ∈ P2(Fq) : η(α0) =−1

}and {

(α0 : 1 : 0) ∈ P2(Fq) : η(α0) =−1}

are denoted by A0, A1 and A2, respectively.

4.3 Rudiments of the Stöhr-Voloch theory

In this section, some notions and basic facts from the Stöhr-Voloch theory are presented.We believe the results here are well known by the specialists. Nevertheless, as some of them arenot explicitly stated in the literature, their proofs are provided.

Definition 4.3.1. For any homogeneous polynomial A(X0,X1,X2) ∈ Fq[X0,X1,X2], consider thepolynomial

Φq(A) := Xq0

∂A∂X0

+Xq1

∂A∂X1

+Xq2

∂A∂X2

. (4.3)

An absolutely irreducible plane curve G : G(X0,X1,X2) = 0 defined over Fq is called Fq-

Frobenius nonclassical if

G |Φq(G). (4.4)

Otherwise, G is called Fq-Frobenius classical.


Note that (4.4) has a geometric meaning: the Frobenius map P ↦→ Pq takes each nonsin-gular point P ∈ G to the tangent line to G at P.

The following result, which is important to prove the absolute irreducibility of F , isbased essentially on (CARLIN; VOLOCH, 2004, Theorem 1).

Theorem 4.3.2. Let G be a plane curve of degree ą defined over Fq. If all absolutely irreduciblecomponents of G defined over Fq are Fq-Frobenius classical, then

#G (Fq)6ą(ą+q−1)

2.

Moreover, if #G (Fq)>ą(ą+q−1)

2 −max{ą−1,2ą−5}, then G is absolutely irreducible.

Proof. Let G1, . . . ,Gl be the (irreducible but not necessarily absolutely irreducible) componentsof G defined over Fq, and let ąi be the degree of Gi. By hypothesis, ąi > 1 for all i (since thecondition ‘all absolutely irreducible components of G defined over Fq are Fq-Frobenius classical’implies in particular that Gi is not a line). If Gi is absolutely irreducible, then Theorem 3.0.1yields

#Gi(Fq)6 ąi(ąi +q−1)/2. (4.5)

On the other hand, from (VILLEGAS; VOLOCH, 1999, Lemma 3.3), if Gi is not absolutelyirreducible, then #Gi(Fq)6 ą2

i /4. Since ą2i /4 < ąi(ąi+q−1)/2, the bound (4.5) is also obtained

in cases where Gi is not absolutely irreducible. Now

#G (Fq)6∑#Gi(Fq)6∑ąi(ąi +q−1)/2. (4.6)

From ∑ąi = ą,

∑ąi(ąi +q−1)/2 = ∑ą(ą+q−1)/2−∑i<k

ąiąk,

which combined with (4.6) gives the first statement of the theorem.

To prove the second statement, suppose that G is not absolutely irreducible. If l > 1, thenassume without loss of generality that ą1 6 ąi, for all i > 1. It follows that

∑i<k

ąiąk > ą1(ą2 + · · ·+ąl)> 2(ą2 + · · ·+ąl)> ą1 + · · ·+ąl = ą

and that

∑i<k

ąiąk > ą1(ą1 + · · ·+ąl)−ą1ą1 = ą1ą−ą21 > 2ą−4,

where the last inequality follows from the fact that ą21−ą1ą+2ą−46 0 for 26 ą1 6 ą−2. If

l = 1, then G is irreducible but not absolutely irreducible and, again by (VILLEGAS; VOLOCH,1999, Lemma 3.3), #G (Fq)6 ą2/4. This implies that #G (Fq)6

ą(ą+q−1)2 −ą unless q = 2 and

ą = 1, case in which G is absolutely irreducible, a contradiction with the hypothesis. Also, thisimplies that #G (Fq)6

ą(ą+q−1)2 − (2ą−4), which completes the proof. �


Finally, the proofs of the nonsingularity and the classicality of F are based on Theorems4.3.3 and 4.3.4, which are slightly adapted versions of Theorem 3.0.1 and (HIRSCHFELD;KORCHMÁROS, 2001, Theorem 1.3), respectively.

Theorem 4.3.3. Let G : G(X0,X1,X2) = 0 be an Fq-Frobenius classical plane curve of degreeą and let P1, . . . , Pk be the distinct inflection points of G defined over Fq. If TPi

is the tangentline to G at Pi and j(i)2 = I(Pi,G ∩TPi

)> 3 is the intersection multiplicity of G and TPiat Pi, for

i = 1, . . . ,k, then

#G (Fq)6ą(ą+q−1)−∑

ki=1( j(i)2 −2)

2. (4.7)

Moreover, if equality holds in (4.7), then G is nonsingular.

Proof. Let H : Φq(G) = 0. Since G is Fq-Frobenius classical, from Bézout’s Theorem (seeTheorem 1.1.36) one may have

∑P∈G∩H

I(P,G ∩H ) = ą(ą+q−1).

Also, from Euler’s Formula,

ą G(X0,X1,X2) = X0∂G∂X0

+X1∂G∂X1

+X2∂G∂X2

(4.8)

and therefore G (Fq)⊆H (Fq), which implies that

∑P∈G (Fq)

I(P,G ∩H )6 ∑P∈G∩H

I(P,G ∩H ) = ą(ą+q−1). (4.9)

In order to complete the proof, the left side of the above inequality needs to be analyzed,which is done in two steps.

Step 1: I(P,G ∩H ) > 2, for all P ∈ G (Fq). Moreover, if P ∈ G (Fq) is a singular point of G ,then the strict inequality holds, that is, I(P,G ∩H )> 2.

Proof of Step 1.

First, for each i ∈ {0,1,2},

∂Φq(G)

∂Xi= qXq−1

i∂G∂Xi

+Xq0

∂ 2G∂X0∂Xi

+Xq1

∂ 2G∂X1∂Xi

+Xq2

∂ 2G∂X2∂Xi

= Xq0

∂ 2G∂Xi∂X0

+Xq1

∂ 2G∂Xi∂X1

+Xq2

∂ 2G∂Xi∂X2

= Φq

(∂G∂Xi

),

and so, if P ∈ G (Fq)⊆H (Fq), then

∂Φq(G)

∂Xi

∣∣∣∣P= (ą−1)

∂G∂Xi

∣∣∣∣P


by Euler’s Formula applied to ∂G∂Xi

. In particular, if P ∈ G (Fq)⊆H (Fq) is a singular point of G ,then P is also a singular point of H , and, by Theorem 1.1.35,

I(P,G ∩H )>mGPm

HP > 2 ·2 = 4 > 2,

where mGP and mH

P are the multiplicities of P at G and H , respectively. Further, if P ∈ G (Fq)⊆H (Fq) is a nonsingular point of G , then either P is a singular point of H (if p | (ą−1)) or P

is a nonsingular point of H (if p - (ą−1)) and the tangent line to H at P coincides with thetangent line to G at P. Therefore, in both circumstances

I(P,G ∩H )> 2,

which completes the proof of Step 1. �

Step 2: I(Pi,G ∩H )> j(i)2 , for all i = 1, . . . ,k.

Proof of Step 2.

Consider i ∈ {1, . . . ,k}. From a particular case of (ARAKELIAN; BORGES, 2017,Lemma 3.3), the result of this step follows provided that I(Pi,H ∩TPi

)> j(i)2 .

In order to simplify the notation, define P := Pi and j2 := j(i)2 , and write TP : L(X0,X1,X2)=

0. Also, without loss of generality, suppose that P = (α0 : α1 : 1), with α0,α1 ∈ Fq.

First, note that if p | ą, then

I(P,H ∩TP) = I(P,H ∩TP),

and if p - ą, then

I(P,H ∩TP)>min{

I(P,G ∩TP), I(P,H ∩TP)

},

where H : H(X0,X1,X2) = (Xq0 −X0Xq−1

2 )∂G∂X0

+(Xq1 −X1Xq−1

2 )∂G∂X1

= 0. Indeed, by Euler’s

Formula (4.8)

Φq(G) = Xq0

∂G∂X0

+Xq1

∂G∂X1

+Xq2

∂G∂X2

= Xq0

∂G∂X0

+Xq1

∂G∂X1

+Xq−12 X2

∂G∂X2

= Xq0

∂G∂X0

+Xq1

∂G∂X1

+Xq−12

(ąG(X0,X1,X2)−X0

∂G∂X0−X1

∂G∂X1

)= (Xq

0 −X0Xq−12 )

∂G∂X0

+(Xq1 −X1Xq−1

2 )∂G∂X1

+ąXq−12 G(X0,X1,X2).

Therefore, one needs to show that

I(P,H ∩TP)> j2.


Writing

G*(X0 +α0,X1 +α1) = G(X0 +α0,X1 +α1,1)

= G1(X0,X1)+G2(X0,X1)+ · · ·+Gą(X0,X1),

where Gl(X0,X1) is a homogeneous polynomial of degree l for each l ∈ {1,2, . . . ,ą}, and

H*(X0,X1) = H(X0,X1,1) = (Xq0 −X0)

∂G*∂X0

+(Xq1 −X1)

∂G*∂X1

,

since α0,α1 ∈ Fq, it follows that

H*(X0 +α0,X1 +α1) = (Xq0 −X0)

∂G1

∂X0+(Xq

1 −X1)∂G1

∂X1+

+ (Xq0 −X0)

∂G2

∂X0+(Xq

1 −X1)∂G2

∂X1+

· · ·

+ (Xq0 −X0)

∂Gą

∂X0+(Xq

1 −X1)∂Gą

∂X1.

Now, applying Euler’s Formula to Gl(X0,X1) for each l ∈ {1,2, . . . ,ą},

H*(X0 +α0,X1 +α1) = −1 ·G1(X0,X1)−2 ·G2(X0,X1)−·· ·−ą ·Gą(X0,X1)+

+ Xq0

∂G1

∂X0+Xq

1∂G1

∂X1+

+ Xq0

∂G2

∂X0+Xq

1∂G2

∂X1+

· · ·

+ Xq0

∂Gą

∂X0+Xq

1∂Gą

∂X1.

By Remark 1.1.33, L*(X0 +α0,X1 +α1) | Gl(X0,X1), for all 1 6 l < j2, and L*(X0 +

α0,X1 +α1) - G j2(X0,X1).

If the homogeneous components of degree l of H*(X0 +α0,X1 +α1) are exactly ofthe form −l ·Gl(X0,X1), for all 1 6 l < j2, then the result follows by applying the aboveconsiderations and Remark 1.1.33 to the polynomial H*(X0 +α0,X1 +α1).

On the other hand, if the homogeneous component of degree l of H*(X0+α0,X1+α1) isof the form−l ·Gl(X0,X1)+Xq

0∂Gl1∂X0

+Xq1

∂Gl1∂X1

for some 16 l < j2, where l1 = l+1−q < l < j2,then writing L*(X0 +α0,X1 +α1) = β0X0 + β1X1, with β0,β1 ∈ Fq (which is possible sinceP = (α0 : α1 : 1) ∈ P2(Fq) is a nonsingular point of G ), and

Gl1(X0,X1) = L*(X0 +α0,X1 +α1)Gl1(X0,X1),


it follows that

Xq0

∂Gl1∂X0

+Xq1

∂Gl1∂X1

= Xq0

(β0Gl1(X0,X1)+L*(X0 +α0,X1 +α1)

∂ Gl1∂X0

)+ Xq

1

(β1Gl1(X0,X1)+L*(X0 +α0,X1 +α1)

∂ Gl1∂X1

)= (β0Xq

0 +β1Xq2 )Gl1(X0,X1)+L*(X0 +α0,X1 +α1)

(Xq

0∂ Gl1∂X0

+Xq1

∂ Gl1∂X1

)= (β0X0 +β1X2)

qGl1(X0,X1)+L*(X0 +α0,X1 +α1)

(Xq

0∂ Gl1∂X0

+Xq1

∂ Gl1∂X1

)is also divisible by L*(X0 +α0,X1 +α1) = β0X0 +β1X1.

Therefore, from Remark 1.1.33,

I(P,H ∩TP)> j2,

and the proof of Step 2 follows. �

From the previous two steps,

∑P∈G (Fq)

I(P,G ∩H ) =k

∑i=1

I(Pi,G ∩H )+ ∑P∈G (Fq)∖{P1,...,Pk}

I(P,G ∩H )

>k

∑i=1

j(i)2 +2 ·#G (Fq)∖{P1, . . . , Pk}

=k

∑i=1

( j(i)2 −2)+2 ·#G (Fq),

and thus from (4.9)

#G (Fq)6ą(ą+q−1)−∑

ki=1( j(i)2 −2)

2.

Assuming equality in (4.7),

k

∑i=1

j(i)2 +2 ·#G (Fq)∖{P1, . . . , Pk}= ą(ą+q−1) = ∑P∈G∩H

I(P,G ∩H ),

and then the following occurs:

1. I(P,G ∩H ) = 2, for all P ∈ G (Fq)∖{P1, . . . , Pk};

2. I(Pi,G ∩H ) = j(i)2 , for all i = 1, . . . ,k;

3. G ∩H = G (Fq).

From the definition of H , it follows that any singular point P ∈ G must be a point ofH and then assertion 3 above gives that P ∈ G (Fq). However, if P ∈ G (Fq) is singular, thenI(P,G ∩H ) > 2 by Step 1, which contradicts assertion 1. Hence, G is a nonsingular planecurve. �

4.4. Points and linear components of curve C 89

Theorem 4.3.4. Let G : G(X0,X1,X2) = 0 be an Fq-Frobenius classical plane curve of degreeą < q. If G has an infinite number of inflection points, then

#G (Fq)6ą(ą+q−1)

pk (4.10)

for some k > 1.

Proof. Considering the notation as in (HIRSCHFELD; KORCHMÁROS, 2001), let Bq be thenumber of branches of G centered at points of P2(Fq). Therefore, since #G (Fq)6 Bq, all that isneeded is to show that for some k > 1

Bq 6(2g−2)+(q+2)ą

pk 6ą(ą+q−1)

pk , (4.11)

where g is the genus of G . Note that the second inequality in (4.11) is trivial. The first one isproved by considering ν = 1 in inequality (3.1) in the proof of (HIRSCHFELD; KORCHMÁROS,2001, Theorem 1.3). In fact, following the notation therein, since G is Fq-Frobenius classical,

vP(S)> vP([x(t)D(1)t y(t)− y(t)D(1)

t x(t)])> r+ s−1> ε,

if vP(S)< rq, andvP(S)> rq > rm> rs> ε

otherwise. Hence, since G has infinitely many inflection points, and then ε = pk for some k > 1(see (GARCIA; VOLOCH, 1987, Proposition 2)), the first inequality in (4.11) follows. �

4.4 Points and linear components of curve C

In this section, the Fq-rational points and linear components of C given in (4.2) areinvestigated.

4.4.1 Points with zero coordinates

The following lemma follows from the definitions.

Lemma 4.4.1. If e0 = e1 = e2 = 0, then a point (α0 : α1 : α2) ∈ P2(Fq) is on C if and only if{η(α0),η(α1),η(α2)}= {−1,0,1}. In particular, #C (Fq) = 3d.

Lemma 4.4.2. For a point P = (α0 : α1 : α2) ∈ P2(Fq) with α0α1α2 = 0, the following occurs.

1. For i = 0,1,2,Pi ∈ C (Fq) if and only if η(ei) =−1;

2. Let i,k ∈ {0,1,2} be distinct elements such that αiαk = 0. Then P ∈ C (Fq) if and only if

ei = ek = 0 and η(αiαk) =−1

orη(−eiek) =−1 and P = Pik.


Proof. Let P = (α0 : α1 : α2) be a point of P2(Fq) with α0α1α2 = 0. Statement 1 follows directlyfrom the definition of C . Now, suppose without loss of generality that α0α1 = 0 and α2 = 0.Then the following assertions are equivalent:

∙ P ∈ C (Fq);

∙ αd0 +αd

1 +(e0α0 + e1α1)d = 0;

∙ e0α0 + e1α1 = 0 and η(α0α1) =−1;

∙ Either e0 = e1 = 0 and η(α0α1) =−1 or η(−e0e1) =−1 and P = P01,

which completes the proof. �

Based on these results, the number of Fq-rational points on C with zero coordinates issummarized in Table 1, for all possible quadratic characters of e0, e1 and e2.

{η(e0),η(e1),η(e2)} i = 2 i = 1 and d odd i = 1 and d even{ 1,1,1 } 0 3 0{ -1,1,1 } 1 1 2{ -1,-1,1 } 2 1 2{ -1,-1,-1 } 3 3 0{ 0,1,1 } 0 1 0{ -1,0,1 } 1 0 1{ -1,-1,0 } 2 1 0{ 0,0,1 } 0 d d{ -1,0,0} 1 d d{ 0,0,0 } 0 3d 3d

Table 1 – The number of Fq-rational points on C with i ∈ {1,2} zero coordinates.

4.4.2 Points without zero coordinates

A point P = (1 : α1 : α2) ∈ P2(Fq), with α1α2 = 0, is on C (Fq) if and only if one of thefollowing cases occurs:

(1) η(α1) = 1, η(α2) =−1 and η(e0 + e1α1 + e2α2) =−1;

(2) η(α1) =−1, η(α2) = 1 and η(e0 + e1α1 + e2α2) =−1;

(3) η(α1) =−1, η(α2) =−1 and η(e0 + e1α1 + e2α2) = 1.

If N(i) is the number of points satisfying case (i), for i ∈ {1,2,3}, then N(1), N(2) andN(3) are the cardinalities of the sets{

(γ21 ,λγ

22 ,λγ

23 ) ∈ F*q

3 : γi ∈ F*q, for i = 1,2,3, and e0 + e1γ21 + e2λγ

22 = λγ

23

},

{(λγ

21 ,γ

22 ,λγ

23 ) ∈ F*q

3 : γi ∈ F*q, for i = 1,2,3, and e0 + e1λγ21 + e2γ

22 = λγ

23

},


and {(λγ

21 ,λγ

22 ,γ

23 ) ∈ F*q

3 : γi ∈ F*q, for i = 1,2,3, and e0 + e1λγ21 + e2λγ

22 = γ

23

},

respectively, where λ is a fixed non-square element of Fq.

Let

N( l

∑k=1

βkY 2k = β

)represent the number of solutions (γ1, . . . ,γl) ∈ Fl

q of the equation

l

∑k=1

βkY 2k = β

defined over Fq, and let

a1 = e1, a2 = λe2, a3 =−λ , α =−e0, if i = 1,

a1 = λe1, a2 = e2, a3 =−λ , α =−e0, if i = 2,

anda1 = λe1, a2 = λe2, a3 =−1, α =−e0, if i = 3.

Then the number N(i), with i ∈ {1,2,3}, is determined by the following expressions, for allpossible cases of e0,e1 and e2.

1. Case e0e1e2 = 0

8N(i) = N(a1Y 21 +a2Y 2

2 +a3Y 23 = α)−

3

∑k=1

N( 3

∑l=1,l =k

alY 2l = α

)

+3

∑k=1

N(akY 2k = α).

2. Case e0ek = 0,ek1 = 0, with k,k1 ∈ {1,2}

8N(i) = (q−1)[

N(akY 2k +a3Y 2

3 = α)−3

∑l=1,l =k1

N(alY 2l = α)

].

3. Case e1e2 = 0,e0 = 0

8N(i) = N(a1Y 21 +a2Y 2

2 +a3Y 23 = α)−

3

∑k=1

N( 3

∑l=1,l =k

alY 2l = α

)

+3

∑k=1

N(akY 2k = α)−1.

4. Case e0 = 0,e1 = e2 = 0

8N(i) = (q−1)2N(a3Y 23 = α).


5. Case ek = 0,e0 = ek1 = 0, with k,k1 ∈ {1,2}

8N(i) = (q−1)[

N(akY 2k +a3Y 2

3 = α)−3

∑l=1,l =k1

N(alY 2l = α)+1

].

Table 2 presents the values of N(1)+N(2)+N(3), which are calculated using the followinglemma given by Propositions 1 and 2 of (JOLY, 1973, Chapter 6).

Lemma 4.4.3. Let p > 2 and l a positive integer. The number of solutions in Flq of the equation

β1Y 21 + · · ·+βlY 2

l = β , with βk ∈ F*q for all k = 1, . . . , l, is given by:

ql−1 +η((−1)l/2β1 · · ·βl)(ql/2−q(l−2)/2), if β = 0 and l ≡ 0 (mod 2),

ql−1−η((−1)l/2β1 · · ·βl)q(l−2)/2, if β = 0 and l ≡ 0 (mod 2),

ql−1, if β = 0 and l ≡ 1 (mod 2),

ql−1 +η((−1)(l−1)/2β1 · · ·βlβ )q(l−1)/2, if β = 0 and l ≡ 1 (mod 2).

e0e1e2 = 0{η(e0),η(e1),η(e2)} d odd d even

{1,1,1} 3(q−1)(q−3)8

3(q−1)2

8

{-1,1,1} 3q2−6q+78

3(q−1)(q−3)8

{-1,-1,1} 3(q−1)(q−3)8

3q2−6q+118

{-1,-1,-1} 3(q2−2q+5)8

3(q−1)(q−3)8

Exactly one of the elements e0,e1,e2 is zero{η(e0),η(e1),η(e2)} d odd d even

{0,1,1} (q−1)(3q−5)8

3(q−1)2

8{-1,0,1} (q−1)(3q−5)

8(q−1)(3q−7)

8{-1,-1,0} 3(q−1)(q−3)

8(q−1)(3q−7)

8Exactly two of the elements e0,e1,e2 are zero

{η(e0),η(e1),η(e2)} d odd d even

{0,0,1} (q−1)2

4(q−1)2

4

{-1,0,0} (q−1)2

2(q−1)2

2Table 2 – The number N(1)+N(2)+N(3).

4.4.3 Linear components

Here, Lemmas 4.4.4 and 4.4.8 provide a description of the linear components of C ,which are summarized, for e0e1e2 = 0, in Table 3.

Lemma 4.4.4. The linear components of C and the circumstances in which they arise are asfollows:

1. The plane curve C is a union of d lines if and only if {η(e0),η(e1),η(e2)}= {−1,0,0};


2. If C is not a union of d lines, then a line L is a component of C if and only if L is givenby eiXi + ekXk = 0 and η(−eiek) = η(el) =−1, with {i,k, l}= {0,1,2}.

To prove the previous result, the following proposition and its corollary need to beconsidered first.

Proposition 4.4.5. Let K be an algebraically closed field of arbitrary characteristic p andconsider G : G(X0,X1,X2) = 0 a nonsingular plane curve in P2(K) of degree ą> 2. Suppose thatp - ą. If k is the number of total inflection points of G , then the surface in P3(K)

S : Xą3 −G(X0,X1,X2) = 0

contains exactly ką lines.

Proof. Let L be a line on S given by the intersection of two distinct hyperplanes. Aftera suitable rotation of the planes around L , one may assume that one of them is given byX2 = α0X0 +α1X1, α0,α1 ∈K. Thus Xą

3 −G(X0,X1,α0X0 +α1X1) has a linear factor, say X3−β0X0−β1X1 ∈ K[X0,X1,X3], corresponding to the other plane. This implies by the Euclideanalgorithm for divisibility of polynomials1 that G(X0,X1,α0X0 +α1X1) = (β0X0 +β1X1)

ą, whichis equivalent to X2 = α0X0 +α1X1 being the tangent to G at some total inflection point P ∈ G .

On the other hand, let P be any total inflection point of G . Without loss of generality, onecan suppose that the tangent line TP to G at P is given by X2 = α0X0 +α1X1, with α0,α1 ∈K.Then, G(X0,X1,α0X0 +α1X1) = (β0X0 +β1X1)

ą, for some β0,β1 ∈K, which gives rise to the ą

lines on S

Li :

{X2 = α0X0 +α1X1

ξiX3 = β0X0 +β1X1,(4.12)

where ξ1, . . . ,ξą are the ą-th roots of unity.

Since distinct total inflection points P ∈ C yield disjoint sets of ą lines, the resultfollows. �

Remark 4.4.6. Suppose that K= Fq and that G is defined over Fq. Also, suppose that Fq hasthe ą-th roots of unity. Then, from the proof of Proposition 4.4.5 and considering the notationtherein, the following assertion holds.

If k is the number of total inflection points of G defined over Fq, then the surface S contains

exactly ką lines defined over Fq.1 Let R be any (commutative) ring and Y i−α,Y l−β ∈ R[Y ], with α,β ∈ R∖{0} and 16 i6 l. Applying

the Euclidian algorithm for divisibility of polynomials (which can be used here since Y i−α is a monicpolynomial), we have that

(Y i−α) | (Y l−β ) ⇔ i | l and αli = β .


Corollary 4.4.7. Let ą> 3 be an integer such that p - ą. If ą = pi +1 for some i> 1, then theFermat surface in P3(Fq)

S : Xą0 +Xą

1 +Xą2 +Xą

3 = 0

has (pi +1)(p3i +1) lines. Otherwise, the number of lines is 3ą2.

Proof. It is well known that the Hermitian curve

H : X pi+10 +X pi+1

1 +X pi+12 = 0

has p3i +1 rational points over Fp2i , and that those are all the total inflection points of H .

We claim that all Fermat curves of degree ą = pi +1 have only the well-known 3ą totalinflection points, namely the points on X0X1X2 = 0.

In fact, let P = (α0 : α1 : 1), with α0α1 = 0, be a total inflection point of

G : Xą0 +Xą

1 +Xą2 = 0,

and let TP : αą−10 X0 +α

ą−11 X1 +X2 = 0 be the tangent line to G at P. Thus, for α =− 1

αą−11

and

β = (α0α1)ą−1 = 0, it follows that α0 is the only zero of

Xą0 +(α−βX0)

ą +1,

and then the only zero of its derivative

ą

(Xą−1

0 −β (α−βX0)ą−1).

The latter is equivalent to (X0

α−βX0

)ą−1

= β

having only one root, which clearly happens if and only if d−1 is a power of p. This proves theclaim.

Now the result follows immediately from Proposition 4.4.5. �

Proof of Lemma 4.4.4. Statements 1 and 2 (⇐) follow from the definition of the plane curve C .

To prove statement 2 (⇒), suppose that C is not a union of d lines, and let L be a linearcomponent of C .

Let us look at C as the plane curve on the Fermat surface S : Xd0 +Xd

1 +Xd2 +Xd

3 = 0 cutout by the hyperplane P : X3− e0X0− e1X1− e2X2, and let L be the corresponding line.

Then, without loss of generality such line L is given by

L :

{X0 = ξ1X1

X3 = ξ2X2,(4.13)

4.5. Preliminary result 95

where ξ1,ξ2 ∈Fq are d-th roots of−1, that is, η(ξi) =−1, for i= 1,2. Note that P and X3 = ξ2X2

are not the same hyperplane as, otherwise, C would be a union of the d lines. Therefore, L is inthe intersection of the following hyperplanes

X0 = ξ1X1

X3 = ξ2X2

X3 = e0X0 + e1X1 + e2X2,

(4.14)

which implies that ξ1 =−e1/e0, ξ2 = e2, and then L must be given by e0X0 + e1X1 = 0. �

Lemma 4.4.8. If e0,e1 and e2 are all nonzero, then the linear components of C have multi-plicity at most 1. That is, none of (e0X0 + e1X1)

2, (e0X0 + e2X2)2 or (e1X1 + e2X2)

2 dividesC(X0,X1,X2).

Proof. Without loss of generality, assume that L01 : e0X0 + e1X1 = 0 is a component of C , andthen η(e2) =−1 by Lemma 4.4.4. Since

C(X0,X1,X2) = Xd0 +Xd

1 +Xd2 +(e0X0 + e1X1)

d +(e2X2)d +

+d−1

∑i=1

(di

)(e0X0 + e1X1)

i(e2X2)d−i,

it follows that (e0X0 + e1X1) | (Xd0 +Xd

1 ). If (e0X0 + e1X1)2 |C(X0,X1,X2) and

(e0X0 + e1X1)2 | (Xd

0 +Xd1 +d(e0X0 + e1X1)(e2X2)

d−1),

then e2 = 0, which is a contradiction. �

(η(e0),η(e1),η(e2)) d odd d even(1,1,1) - -(1,1,−1) e0X0 + e1X1 -(1,−1,1) e0X0 + e2X2 -(−1,1,1) e1X1 + e2X2 -(1,−1,−1) - e0X0 + e1X1,e0X0 + e2X2(−1,1,−1) - e0X0 + e1X1,e1X1 + e2X2(−1,−1,1) - e0X0 + e2X2,e1X1 + e2X2(−1,−1,−1) e0X0 + e1X1,e0X0 + e2X2,e1X1 + e2X2 -

Table 3 – Linear components of C for e0e1e2 = 0.

4.5 Preliminary result

Theorem 4.5.1. Let C be the plane curve given in (4.2). The union of the linear components ofC is Fq-disjoint from its remaining components.


Proof. Let n ∈ {0,1,2,3,d} be the number of linear components of C . If n ∈ {0,d}, then theproof is complete. Now, each of the remaining cases is considered separately.

Case n = 1

Without loss of generality, let L01 : e0X0+e1X1 = 0 be the linear component of C . Then,d is odd, η(e0) = η(e1) = 1, and η(e2) = η(−e0e1) =−1 (see Table 3).

Assume that the point P = (α0 : α1 : α2) ∈ P2(Fq) lies on L01 and on an additionalcomponent of C . Then the polynomial

R(X1) = αd0 +Xd

1 +αd2 +(e0α0 + e1X1 + e2α2)

d

vanishes at α1 with multiplicity at least two. Therefore,

dRdX1

(α1) = dαd−11 +de1(e2α2)

d−1 = 0,

that is, αd−11 + e1(e2α2)

d−1 = 0.

If α1 = 0, then α0 = α2 = 0. Since this cannot occur, it follows that α1 = 0 and

e1

(e2α2

α1

)d−1

=−1.

This implies that η(−1) = 1, which contradicts the fact that d is odd. Indeed,

e1

(e2α2

α1

)d−1

= e1

(e2α2

α1

)d(α1

e2α2

)=−1.

Therefore, if η(α1) = η(α2), then η

(e2α2

α1

)=

(e2α2

α1

)d

=−1 and

η(−1) = η

(− e1

(α1

e2α2

))= η(e1) = 1.

Further, if η(α1) = η(α2), then η

(e2α2

α1

)=

(e2α2

α1

)d

= 1 and

η(−1) = η

(e1

(α1

e2α2

))= η(e1) = 1.

�

Case n = 2

Without loss of generality, let L01 : e0X0 + e1X1 = 0 and L02 : e0X0 + e2X2 = 0 be thelinear components of C . Then d is even, η(e0) = 1, and η(e1) = η(e2) =−1 (see Table 3).

Note that if d = 2 (which occurs if and only if q = 5), then the proof is complete.Therefore, hereafter let us consider d > 2.

4.5. Preliminary result 97

Assume that P = (α0 : α1 : α2) ∈ P2(Fq) lies on L01 and on an additional component ofC , but not on L02. Then the polynomial

R(X1) = αd0 +Xd

1 +αd2 +(e0α0 + e1X1 + e2α2)

d

vanishes at α1 with multiplicity at least two and, therefore,

dRdX1

(α1) = dαd−11 +de1(e2α2)

d−1 = 0,


d−1 = 0.

If α1 = 0, then α0 = α2 = 0. Since this is impossible, it follows that α1 = 0 and so

e1

(e2α2

α1

)d−1

= e1

(e2α2

α1

)d(α1

e2α2

)=−1,

which implies that

e1η(α1α2) =e2α2

α1.

If η(α1α2) = 1, then e1α1 = e2α2 and P lies on L02, which is a contradiction. On theother hand, if η(α1α2) =−1, then e1 =−e2α2

α1and η(e1) = 1, which is also a contradiction since

η(e1) =−1.

Analogously, one can show that P = (α0 : α1 : α2) ∈ P2(Fq) lying on L02 and on anadditional component of C , but not on L01 leads to a contradiction.

Now, assume that the point P ∈L01∩L02 lies on an additional component of C . Then

R(X0) = Xd0 +α

d1 +α

d2 +(e0X0 + e1α1 + e2α2)

d

vanishes at α0 with multiplicity at least three. Therefore,

d2RdX2

0(α0) = d(d−1)αd−2

0 +d(d−1)e20(e2α2)

d−2 = 0,

that is, αd−20 +e2

0(e2α2)d−2 = 0. However, since e0α0+e2α2 = 0, d is even and η(e0) = 1, it fol-

lows that 2αd−20 = 0, that is, α0 =α1 =α2 = 0, which is impossible. �

Case n = 3

Suppose that C has as linear components the lines L01 : e0X0 + e1X1 = 0, L02 : e0X0 +

e2X2 = 0, and L12 : e1X1+e2X2 = 0. Then η(e0) = η(e1) = η(e2) =−1 and d is odd (see Table3).

Assume that P = (α0 : α1 : α2) ∈ P2(Fq) lies on L01 and on an additional component ofC , but not on L02 and L12. Thus

R(X1) = αd0 +Xd

1 +αd2 +(e0α0 + e1X1 + e2α2)

d


vanishes at α1 with multiplicity at least two. Therefore,

dRdX1

(α1) = dαd−11 +de1(e2α2)

d−1 = 0,


d−1 = 0.

If α1 = 0, then α0 = α2 = 0. Since this cannot occur, it follows that α1 = 0 and, as inthe previous case, it is possible to write

e1η(α1α2) =e2α2

α1.

If η(α1α2) = 1, then e1α1 = e2α2 and P ∈L02, which is a contradiction. If η(α1α2) =

−1, then e1α1 + e2α2 = 0 and P ∈L12, which is also a contradiction.

Analogously, one can show that P = (α0 : α1 : α2) ∈ P2(Fq) lying on L02 and on anadditional component of C , but not on L01 and L12, and P = (α0 : α1 : α2) ∈ P2(Fq) lyingon L12 and on an additional component of C , but not on L01 and L02, lead us also to acontradiction.

Finally, without loss of generality, assume that P ∈L01∩L02 (and then P /∈L12) lieson an additional component of C . Then

R(X0) = Xd0 +α

d1 +α

d2 +(e0X0 + e1α1 + e2α2)

d

vanishes at α0 with multiplicity at least three. Therefore,

d2RdX2

0(α0) = d(d−1)αd−2

0 +d(d−1)e20(e2α2)

d−2 = 0,

that is, αd−20 + e2

0(e2α2)d−2 = 0. However, since e0α0 + e2α2 = 0, d is odd (and then d−2> 1)

and η(e0) =−1, it follows that 2αd−20 = 0, that is, α0 = α1 = α2 = 0, which is impossible. �

Hence, for all possible cases, it has been shown that assuming P ∈ P2(Fq) to be apoint on a linear component of C and on an additional nonlinear component of C leads to acontradiction. �

4.6 Main resultBefore proving the central result of this work in Subsection 4.6.2, the absolute irreducibil-

ity of F needs to be studied. This is done in Subsection 4.6.1.

4.6.1 Frobenius classicality and absolute irreducibility

Let F be the union of the nonlinear components of C . Our objective here is to show thatF consists of only one absolutely irreducible nonlinear component.

4.6. Main result 99

In cases where {η(e0),η(e1),η(e2)} is either {0,0,0} or {0,0,1}, F = C is a Fermatcurve, for which the absolute irreducibility is well known.

The study of the remaining cases is centered around Theorem 4.3.2. However, in order toapply it, the following result is necessary. Note that the notation here is given as in Definition4.3.1.

Theorem 4.6.1. Each nonlinear absolutely irreducible component G ⊆ C defined over Fq is anFq-Frobenius classical plane curve.

Proof. Suppose that G is given by

G(X0,X1,X2) = 0,

where G(X0,X1,X2)∈ Fq[X0,X1,X2] is an absolutely irreducible homogeneous polynomial. Then,it is necessary to show that G -Φq(G).

Let H(X0,X1,X2) ∈ Fq[X0,X1,X2] be such that

C(X0,X1,X2) = G(X0,X1,X2)H(X0,X1,X2)

and note thatΦq(C) = GΦq(H)+Φq(G)H.

Therefore, it suffices to prove that G -Φq(C).

Since C(X0,X1,X2) = Xd0 +Xd

1 +Xd2 +Xd

3 , with X3 = e0X0 + e1X1 + e2X2 and d = (q−1)/2, it follows that

Φq(C) = d(X3d0 +X3d

1 +X3d2 +X3d

3 ).

Indeed,

Φq(C) = Xq0 [dXd−1

0 +de0(e0X0 + e1X1 + e2X2)d−1]+

+ Xq1 [dXd−1

1 +de1(e0X0 + e1X1 + e2X2)d−1]+

+ Xq2 [dXd−1

2 +de2(e0X0 + e1X1 + e2X2)d−1]

= d(X3d0 +X3d

1 +X3d2 )+d(e0X0 + e1X1 + e2X2)

d−1(e0Xq0 + e1Xq

1 + e2Xq2 )

= d(X3d0 +X3d

1 +X3d2 )+d(e0X0 + e1X1 + e2X2)

d−1(e0X0 + e1X1 + e2X2)q

= d(X3d0 +X3d

1 +X3d2 )+d(e0X0 + e1X1 + e2X2)

3d

= d(X3d0 +X3d

1 +X3d2 +X3d

3 ).

Finally, from the polynomial identity

X3d0 +X3d

1 +X3d2 +X3d

3 = C(X0,X1,X2)3−3Xd

3 (Xd0 +Xd

1 +Xd2 )C(X0,X1,X2)−

− 3(Xd0 +Xd

1 )(Xd0 +Xd

2 )(Xd1 +Xd

2 ),

it follows that if G |Φq(C), then G | (Xd0 +Xd

1 )(Xd0 +Xd

2 )(Xd1 +Xd

2 ), which contradicts deg(G)>

1. Hence, G is Fq-Frobenius classical. �


Now, if d is the degree of F , based on Tables 4 and 5,

#F (Fq)>d(d+q−1)

2−max{d−1,2d−5},

if and only if at least two of e0,e1 and e2 are nonzero, which includes all cases where

#F (Fq) =12

d(d+q−1)− 12(d−2).

Therefore, in these cases, F is absolutely irreducible by Theorem 4.3.2.

4.6.2 Proof of Theorem 4.1.1

Given a specific set {η(e0),η(e1),η(e2)}, the number of Fq-rational points with zerocoordinates on C , denoted by N(0), is listed in Table 1, and the number N(1)+N(2)+N(3) of Fq-rational points without zero coordinates on C is determined in Subsection 4.4.2 and summarizedin Table 2. Hence,

#C (Fq) = N(0)+N(1)+N(2)+N(3).

By Subsection 4.4.3, C has:

∙ exactly one linear component, if d is odd and {η(e0),η(e1),η(e2)}= {−1,1,1}, in whichcase #F (Fq) = #C (Fq)− (q+1) and d = (q−3)/2;

∙ exactly two linear components, if d is even and {η(e0),η(e1),η(e2)} = {−1,−1,1}, inwhich case #F (Fq) = #C (Fq)− (2q+1) and d = (q−5)/2;

∙ exactly three linear components, if d is odd and {η(e0),η(e1),η(e2)}= {−1,−1,−1}, inwhich case #F (Fq) = #C (Fq)−3q and d = (q−7)/2.

In all other cases, C does not have a linear component, #F (Fq) = #C (Fq), and d =

(q−1)/2.

Tables 4 and 5 summarize the number n of linear components of C , the degree d of F ,and the value of #F (Fq) for the cases in which d is odd and even, respectively. Note that rows(1) and (4) in Table 4 and rows (2) and (5) in Table 5 present two classes of plane curves. Asimple check shows that two plane curves arising from any of the two classes in a particular roware projectively equivalent.

4.6. Main result 101

{η(e0),η(e1),η(e2)} n d #F (Fq)

(1) {1,1,1},{−1,−1,1} 0 (q−1)/2 d(d+q−1)/2−3(d−2)/2(2) {−1,1,1} 1 (q−3)/2 d(d+q−1)/2(3) {−1,−1,−1} 3 (q−7)/2 d(d+q−1)/2(4) {0,1,1},{−1,0,1} 0 (q−1)/2 d(d+q−1)/2− (d−2)/2(5) {−1,−1,0} 0 (q−1)/2 d(d+q−1)/2−3(d−2)/2(6) {0,0,1} 0 (q−1)/2 d(d+q−1)/2−d(d−2)/2(7) {−1,0,0} d - -(8) {0,0,0} 0 (q−1)/2 (d+q−1)/2−3d(d−2)/2

Table 4 – Curve F for d odd.

{η(e0),η(e1),η(e2)} n d #F (Fq)

(1) {1,1,1} 0 (q−1)/2 d(d+q−1)/2(2) {−1,1,1},{−1,−1,−1} 0 (q−1)/2 d(d+q−1)/2−3(d−2)/2(3) {−1,−1,1} 2 (q−5)/2 d(d+q−1)/2(4) {0,1,1} 0 (q−1)/2 d(d+q−1)/2(5) {−1,0,1},{−1,−1,0} 0 (q−1)/2 d(d+q−1)/2− (d−2)(6) {0,0,1} 0 (q−1)/2 d(d+q−1)/2−d(d−2)/2(7) {−1,0,0} d - -(8) {0,0,0} 0 (q−1)/2 d(d+q−1)/2−3d(d−2)/2

Table 5 – Curve F for d even.

The first part of statement 2 follows directly from the results in Tables 4 and 5.

Now the nonsingularity and the classicality of F are discussed. First, recall that theabsolute irreducibility of F was justified in Subsection 4.6.1. Therefore, the idea is to show thatF attains the Stöhr-Voloch bound (4.7) (proving in particular the second part of statement 2)and consequently show that it is nonsingular by Theorem 4.3.3.

If

#F (Fq) =12

d(d+q−1),

then the proof is complete.

In all other cases, F = C and a direct calculation using the information of Tables 6and 7 shows that, for an Fq-rational point P with zero coordinates on C and tangent line TP,the intersection multiplicity of C and TP at P is I(P,C ∩TP) = d. It follows immediately fromTheorem 4.3.3 and Tables 4 and 5 that the Fq-rational points with zero coordinates on C areexactly its (total) Fq-inflection points and that the Stöhr-Voloch bound (4.7) is attained. Hence,F is nonsingular.

Finally, the classicality of F is an immediate consequence of Theorem 4.3.4, since

#F (Fq)>d(d+q−1)

pk


for every k > 1 and, by the considerations in Subsection 4.6.1, F is an Fq-Frobenius classicalplane curve of degree d < q. �

4.6. Main result 103

{η(e0),η(e1),η(e2)} Points Tangent lines

η(e0) = 1,η(e1) = 1,η(e2) = 1P01 ed−1

1 X0 + ed−10 X1 = 0

P02 ed−12 X0 + ed−1

0 X2 = 0P12 ed−1

2 X1 + ed−11 X2 = 0

η(ei) =−1,η(ek) =−1,η(el) = 1Pi eked−1

i Xk + eled−1i Xl = 0

Pk eied−1k Xi + eled−1

k Xl = 0Pik ed−1

k Xi + ed−1i Xk = 0

η(ei) = 0,η(ek) = 1,η(el) = 1 Pkl ed−1l Xk + ed−1

k Xl = 0η(ei) =−1,η(ek) = 0,η(el) = 1 Pi Xl = 0

η(ei) =−1,η(ek) =−1,η(el) = 0Pi Xk = 0Pk Xi = 0Pik ed−1

k Xi + ed−1i Xk = 0

η(ei) = 0,η(ek) = 0,η(el) = 1 Set Al {αd−1i Xi +Xk = 0 : η(αi) =−1}

η(e0) = 0,η(e1) = 0,η(e2) = 0Set A0 {αd−1

1 X1 +X2 = 0 : η(α1) =−1}Set A1 {αd−1

0 X0 +X2 = 0 : η(α0) =−1}Set A2 {αd−1

0 X0 +X1 = 0 : η(α0) =−1}Table 6 – Fq-points P with zero coordinates on C with their respective tangent lines, for d odd and

{η(ei),η(ek),η(el)}= {η(e0),η(e1),η(e2)}, only in cases where #F (Fq)<12d(d+q−1).

{η(e0),η(e1),η(e2)} Points Tangent lines

η(ei) =−1,η(ek) = 1,η(el) = 1Pi eked−1

i Xk + eled−1i Xl = 0

Pik ed−1k Xi− ed−1

i Xk = 0Pil ed−1

l Xi− ed−1i Xl = 0

η(e0) =−1,η(e1) =−1,η(e2) =−1P0 e1ed−1

0 X1 + e2ed−10 X2 = 0

P1 e0ed−11 X0 + e2ed−1

1 X2 = 0P2 e0ed−1

2 X0 + e1ed−12 X1 = 0

η(ei) =−1,η(ek) = 0,η(el) = 1Pi Xl = 0Pil ed−1

l Xi− ed−1i Xl = 0

η(ei) =−1,η(ek) =−1,η(el) = 0Pi Xk = 0Pk Xi = 0

η(ei) = 0,η(ek) = 0,η(el) = 1 Set Al {αd−1i Xi +Xk = 0 : η(αi) =−1}

η(e0) = 0,η(e1) = 0,η(e2) = 0Set A0 {αd−1

1 X1 +X2 = 0 : η(α1) =−1}Set A1 {αd−1

0 X0 +X2 = 0 : η(α0) =−1}Set A2 {αd−1

0 X0 +X1 = 0 : η(α0) =−1}Table 7 – Fq-points P with zero coordinates on C with their respective tangent lines, for d even and

{η(ei),η(ek),η(el)}= {η(e0),η(e1),η(e2)}, only in cases where #F (Fq)<12d(d+q−1).

105

CHAPTER

5ON SOME GENERALIZED FERMAT CURVESAND CHORDS OF AN AFFINELY REGULAR

POLYGON INSCRIBED IN A HYPERBOLA

The present chapter is an extended and adapted version of (BORGES; COUTINHO,2019).

5.1 IntroductionLet Fq be the finite field with q = pm elements, where p is a prime number, and let Y

be a nonsingular geometrically irreducible curve of genus g defined over Fq. A fundamentalproblem in the theory of curves over finite fields is estimating the number Nq(Y ) of Fq-rationalpoints on Y . Apart from a few classes of curves (see (MOISIO, 2007) and (ARAKELIAN;BORGES, 2017) for instance), there is usually no explicit formula for Nq(Y ). Nevertheless,some effective upper bounds for this number can be found in the literature. A famous example,as introduced in Chapter 2, is the Hasse-Weil bound

Nq(Y )6 q+1+2gq1/2. (5.1)

Another noteworthy approach to bound Nq(Y ) was established by Stöhr and Voloch in1986. Their method, more geometric in nature, provides bounds that are dependent on the choiceof an embedding of the curve in some PM(Fq), and which improve (5.1) in several circumstances(see (GARCIA; VOLOCH, 1988) for instance). For more details on the Stöhr-Voloch theory, seealso Chapter 3.

For a,b ∈ Fq satisfying ab /∈ {0,1}, let F be the plane curve with affine equation givenby

F(X ,Y ) = aXdY d−Xd−Y d +b = 0. (5.2)

106 Chapter 5. On the number of rational points on some generalized Fermat curves

This provides an example of a generalized Fermat curve, having been studied from the point ofview of its automorphism group in (ARAKELIAN; SPEZIALI, 2017).

The number Nq(X ) of Fq-rational points on the nonsingular model X defined overFq of F was first investigated in the context of finite geometry to study the number of chordsof an affinely regular polygon in A2(Fq) passing through a given point (see (ABATANGELO;KORCHMÁROS, 1997), (GIULIETTI, 2008)). More precisely, using the Stöhr-Voloch theory,in (ABATANGELO; KORCHMÁROS, 1997) Abatangelo and Korchmáros provided an upperbound for Nq(X ) based on the choice of an embedding of X in P5(Fq), and then on thestudy of its Fq-Frobenius (non)classicality. Later on, Giulietti remarkably improved Abatangeloand Korchmáros’ bound by considering a suitable embedding of X in P3(Fq) (GIULIETTI,2008). Further, more recently, and also using the Stöhr-Voloch theory, the number Nq(X ) wasinvestigated in (FANALI; GIULIETTI, 2012) in the context of generalized Fermat curves.

Accordingly, in terms of the Stöhr-Voloch theory, for Fq(X ) = Fq(x,y), where x,y

satisfy the polynomial equation given by F , and s ∈ {2, . . . ,d− 1}, in this chapter an upperbound for Nq(X ) with respect to the nondegenerate morphism

(· · · : xiyk : · · ·) : X → PN(Fq), (5.3)

with i,k integers satisfying 06 i,k 6 s−1 and 06 i+ k 6 s, is determined in cases where it isFq-Frobenius classical.

Further, based on techniques developed by Garcia and Voloch in (GARCIA; VOLOCH,1988, Section 3) and improved by Mattarei in (MATTAREI, 2007), the attention is focused onthe case where q = p is prime. In particular, if d = (p−1)/n> 3 is a proper divisor of p−1, andp6 d4/4, for an affinely regular n-gon inscribed in a hyperbola, it is obtained that the numbern(a,b) of its chords passing through (a,b) ∈ A2(Fq) is bounded roughly by

3 · (2−1 ·n)2/3, (5.4)

which improves in several cases the upper bound for n(a,b) given in (GIULIETTI, 2008, Theorem4.1):

n(a,b) 6n+1

3. (5.5)

This chapter is organized as follows. In Section 5.3, the number Nq(X ) is studied. Foreach s ∈ {2, . . . ,d−1}, an upper bound for Nq(X ) is provided in cases where the morphism(5.3) is Fq-Frobenius classical. In Subsection 5.3.1, the particular case in which Fq is the primefield Fp is addressed. Finally, in Subsection 5.4.1 some aspects of the theory of affinely regularpolygons are established, with the bound (5.4) being presented in Subsection 5.4.2.

5.2 NotationThe following notation is used throughout this chapter.

5.3. The curve F 107

∙ Here, for an arbitrary field K, K∖{0} is denoted by K*.

∙ P1 and P2 are the points (1 : 0 : 0) and (0 : 1 : 0) of P2(Fq), respectively.

∙ Unless otherwise stated, a curve denotes a geometrically irreducible curve.

∙ For a nonsingular curve Y defined over Fq, Fq(Y ) is its function field, Fq(Y ) is itsFq-rational function field, Y (Fq) is the set of its Fq-rational points, and Nq(Y ) is itsnumber of Fq-rational points.

5.3 The curve F

The following result, which provides basic information about F , can be found in(ABATANGELO; KORCHMÁROS, 1997, Proposition 3.3).

Proposition 5.3.1. Let d be a divisor of q−1. Then, the following holds:

1. The curve F is geometrically irreducible.

2. The genus of F is (d−1)2.

3. The only singular points of F are P1 and P2, and each singularity is ordinary withmultiplicity d. Also, the tangent lines to F at P1 and P2 are given by the affine equationsY = α and X = α , respectively, where αd = a−1, and those tangent lines intersect F atthe corresponding points with multiplicity 2d.

4. The intersection multiplicity of a branch centered at P1 or P2 with its tangent line is d +1.

5. The points (β : 0 : 1) and (0 : β : 1), with β d = b, are inflection points of F . Further, thetangent lines to F at (β : 0 : 1) and (0 : β : 1) are given by the affine equations X = β andY = β , respectively, and those tangent lines intersect F at the corresponding points withmultiplicity d.

For each s ∈ {2, . . . ,d−1}, consider the base-point-free linear series DP1,P2s ⊂ Div(X )

corresponding to the nondegenerate morphism given in (5.3). In other words, let DP1,P2s be the

base-point-free linear series cut out on F by all plane curves (not necessarily irreducible) ofdegree s passing through the singularities P1 and P2 of F .

The following proposition presents some important facts related to the linear seriesDP1,P2

s .

Proposition 5.3.2. For each s ∈ {2, . . . ,d−1}, the following occurs:

1. The linear series DP1,P2s has dimension N =

(s+22

)−3 and degree d= 2d · (s−1).

2. For Q∈X corresponding to a point of F of the form (β : 0 : 1) or (0 : β : 1), with β d = b,the (DP1,P2

s ,Q)-order sequence is given by the elements of{i+ kd : 06 i,k 6 s−1 and 06 i+ k 6 s

}.


3. For Qb ∈X corresponding to a branch b of F centered at P1 or P2, the (DP1,P2s ,Qb)-order

sequence is given by the elements of{i+ k(d +1)−1 : 06 i,k and 06 i+ k 6 s

}∖{−1,s(d +1)−1

}.

Proof. Considering the definitions given in Subsection 1.2.10, for each s ∈ {2, . . . ,d−1}, letΣ

P1,P2s be the linear system of all plane curves (not necessarily irreducible) of degree s passing

through the singularities P1 and P2 of F , and let

DsP1,P2 =

{G ∙F : G ∈ Σ

P1,P2s

}be the linear series cut out on F by the linear system Σ

P1,P2s . Since Σ

P1,P2s has dimension

(s+22

)−3,

and s < 2d, DsP1,P2 has also dimension equal to

(s+22

)− 3 by Proposition 1.2.73. Further, by

Bézout’s Theorem (see Theorem 1.1.36), DsP1,P2 has degree 2ds.

The base locus of DsP1,P2 is the divisor

L∞ ∙F = Q(1)1 + · · ·+Q(1)

d +Q(2)1 + · · ·+Q(2)

d

of degree 2d, where, for i = 1,2 and k = 1, . . . ,d, the Q(i)k ’s are all the distinct points on X for

which the corresponding branches are centered at Pi, and L∞ : Z = 0.

Therefore,

DP1,P2s = Ds

P1,P2−L∞ ∙F =

{D−L∞ ∙F : D ∈ Ds

P1,P2

}(5.6)

has dimension(s+2

2

)−3 and degree 2ds−2d = 2d · (s−1), which proves statement 1.

To prove statement 2, let Q ∈X be a point corresponding to P ∈F , where P is equal to(β : 0 : 1) or (0 : β : 1), with β d = b, and for i = 1,2, let Li be the line passing through P andPi. One can verify that none of the lines L1 and L2 is equal to L∞, and exactly one of themis the tangent line to F at P. Further, from Proposition 5.3.1, the (D1,Q)-order sequence is(0,1,d), where D1 is the base-point-free linear series cut out on F by the linear system of lines.Therefore, considering the reducible plane curves C given by the union of s lines chosen (withmultiplicity) in the set {L1,L2,L∞}, the (DP1,P2

s ,Q)-order sequence is given by the elements of{i+ kd : 06 i,k 6 s−1 and 06 i+ k 6 s

}.

Finally, if Qb ∈X corresponds to a branch b of F centered at P1 or P2, from Proposition5.3.1, the (D1,Qb)-order sequence is (0,1,d +1). Thus, considering the reducible plane curvesC passing through P1 and P2, and given by the union of s lines (possibly chosen with multiplicity),equation (5.6) shows that the (DP1,P2

s ,Qb)-order sequence is comprised by the elements of{i+ k(d +1)−1 : 06 i,k and 06 i+ k 6 s

}∖{−1,s(d +1)−1

},



Based on Theorem 3.2.12 and Proposition 5.3.2, the following result provides an upperbound for the number Nq(X ) in cases where DP1,P2

s is Fq-Frobenius classical.

Corollary 5.3.3. Let s ∈ {2, . . . ,d−1}. If DP1,P2s is Fq-Frobenius classical, then

Nq(X ) 6 (N−1) · (d2−2d)+d · (q+N)

N−2 · d1 ·ϒ1 +d2 ·ϒ2 +d ·ϒ

N, (5.7)

where

∙ d1 and d2 are the number of roots in Fq of the polynomials T d−b and T d−a−1, respec-tively;

∙ ϒ1 = 1+(s−1) ·d−N;

∙ ϒ2 = (s−1) · (d +1)−N;

∙ ϒ = 2(d +1)− s · (4d +3)−N · (N−1)+(s · (2d +3)−3) · N+33 .

Proof. The part

(N−1) · (d2−2d)+d · (q+N)

N

of (5.7) follows directly from Theorem 3.2.12, since DP1,P2s is Fq-Frobenius classical and the

genus of F is (d−1)2 by Proposition 5.3.1.

Further, let Q∈X be a point corresponding to (β : 0 : 1) or (0 : β : 1)∈F , with β d = b,and let Qb ∈X be a point corresponding to a branch b of F centered at P1 or P2. Using thesame notation of Theorem 3.2.12, from Proposition 5.3.2, the numbers A(Q) and A(Qb) aregiven by the following expressions:

A(Q) =

s · (d +1) · −6+(s+1) · (s+2)

6− N · (N−1)

2−N, if Q ∈X (Fq)

s · (d +1) · −6+(s+1) · (s+2)6

− N · (N−1)2

− (1+(s−1) ·d), otherwise

(5.8)

(5.9)

and

A(Qb) =

2− s · (d +1)+(s · (d +2)−3) · (s+1)·(s+2)

6 − N(N−1)2

−N, if Qb ∈X (Fq)

2− s · (d +1)+(s · (d +2)−3) · (s+1)·(s+2)6 − N(N−1)

2

−(s−1) · (d +1), otherwise

(5.10)

(5.11)

since

N

∑i=1

ji(Q) = s · (d +1) · −6+(s+1) · (s+2)6

with jN(Q) = 1+(s−1) ·d,

and

N

∑i=1

ji(Qb) = 2− s · (d +1)+(s · (d +2)−3) · (s+1) · (s+2)6

with jN(Qb) = (s−1) · (d +1).


Therefore,

Nq(X ) 6 (N−1) · (d2−2d)+d · (q+N)

N

−2d1 ·A(Q)(5.8)+2(d−d1) ·A(Q)(5.9)+2d2 ·A(Qb)(5.10)+2(d−d2) ·A(Qb)

(5.11)

N

= (N−1) · (d2−2d)+d · (q+N)

N−2 · d1 ·ϒ1 +d2 ·ϒ2 +d ·ϒ

N,

where

∙ d1 and d2 are the number of roots in Fq of the polynomials T d−b and T d−a−1, respec-tively;

∙ ϒ1 = A(Q)(5.8)−A(Q)(5.9) = 1+(s−1) ·d−N;

∙ ϒ2 = A(Qb)(5.10)−A(Qb)

(5.11) = (s−1) · (d +1)−N;

∙ ϒ = A(Q)(5.9)+A(Qb)(5.11) = 2(d+1)−s ·(4d+3)−N ·(N−1)+(s ·(2d+3)−3) · N+3

3 .

�

5.3.1 The case q = p

From now on, let Fq be the prime field Fp. Our purpose here is to prove the followingtheorem, which is the main result of this chapter.

Theorem 5.3.4. Let d > 3 be a proper divisor of p− 1 and n = (p− 1)/d. If p− 1 6 d ·((d+3)·(d+4)·(3d−1)

12 −3)

, then

Np(X )6 d2 ·(

3 · (21/2 ·n)2/3− 10319· (21/2 ·n)1/3 +

133

).

However, before proving it, some lemmas, which are presented in Subsection 5.3.1.1,need to be introduced.

5.3.1.1 Preliminaries

For each u ∈ [2,+∞) and t0 ∈ [6,+∞), let us define

fu : [6,+∞) → Rt ↦→ 3t2−23t+26

6 +4 · u+3t

,

and

nt0 :=t0 · (t0 +1) · (3t0−10)

12−3 =

14

t30−

712

t20−

56

t0−3.

Lemma 5.3.5. The following occurs:

1. For u ∈ [2,+∞), fu is continuous in [6,+∞), and has derivatives of all orders in (6,+∞).


2. Let t0 ∈ [6,+∞). For u ∈ [2,+∞),

u6 nt0 if and only if fu(t0)6 fu(t0 +1).

3. Fixed u ∈ [n6,+∞),

a) There is only one t0 ∈ [6,+∞) such that u = nt0 .

b) fu has a local minimum at some tmin ∈ (t0, t0 +1). Moreover: since fu is decreasingin [6, tmin] and increasing in [tmin,+∞), tmin is the global minimum point of fu.

Proof. The proof of item 1 is straightforward. Also, item 2 follows from the following equiva-lences

fu(t0)6 fu(t0 +1) ⇔3t2

0−23t0 +266

+4 · u+3t06

3(t0 +1)2−23(t0 +1)+266

+4 · u+3t0 +1

⇔ 4 · u+3t0−4 · u+3

t0 +16

3(t0 +1)2−23(t0 +1)+266

−3t2

0−23t0 +266

⇔ 4 · u+3t0 · (t0 +1)

66t0 +3−23

6

⇔ u+36t0 · (t0 +1) · (3t0−10)

12.

In order to prove item 3-a), let us consider the function

f : [6,+∞) → Rt ↦→ t·(t+1)·(3t−10)

12 −3 = 14t3− 7

12t2− 56t−3

.

Sincedfdt(t) =

34

t2− 76

t− 56> 0

for all t ∈ (6,+∞), this function is increasing and thus a bijection over its image [n6,+∞). Inparticular, for each u ∈ [n6,+∞), there is only one t0 ∈ [6,+∞) such that u = nt0 .

Now, to prove item 3-b), let us fix u ∈ [n6,+∞) and let t0 ∈ [6,+∞) be such that u = nt0 .Then, by item 2, fu(t0) = fu(t0 +1), and thus, by Rolle’s Theorem

dfudt (tmin) = 0

for some tmin ∈ (t0, t0 +1). Since

d2fudt2 (tmin) = 1+8 · u+3

t3min

> 0,

tmin is a local minimum point of fu. Also, from

dfudt (t) =

6t−236−4 · u+3

t2


for all t ∈ (6,+∞), it follows thatdfudt (t)< 0

for all t ∈ (6, tmin), anddfudt (t)> 0

for all t ∈ (tmin,+∞). Indeed,

dfudt (t) = 0 ⇔ t3− 23

6t2−4(u+3) = 0

⇔ t = tmin,

as the discriminant ∆(u) =−432u2− 9431827 u− 59326

9 of the polynomial T 3− 236 T 2−4(u+3) is

negative. Hence, the statement follows from the continuity of dfudt in (6,+∞), since t0 > 6 implies

that

dfudt (t0) =

6t0−236

−4 ·(

312

t0−7

12− 10

12· 1t0

)=−46

12+

2812

+4012· 1t0

=−18t0 +40

12t0< 0,

and

dfudt (t0 +1) =

6(t0 +1)−236

−4 ·(

3t30−7t2

0−10t0

12(t0 +1)2

)=

12(t0 +1)3−46(t0 +1)2−4 · (3t30−7t2

0−10t0)

12(t0 +1)2

=18t2

0−16t0−3412(t0 +1)2

> 0.

Therefore, fu is decreasing in [6, tmin] and increasing in [tmin,+∞), which gives in partic-ular that tmin is a global minimum point of fu. This completes the proof. �

Lemma 5.3.6. Let t0 ∈ Z∩ [6,+∞). If nt0 6 u6 nt0+1, then

min{

fu(t) : t ∈ Z∩ [6,+∞)

}= fu(t0 +1). (5.12)

Furthermore, if 26 u < n6, then

min{

fu(t) : t ∈ Z∩ [6,+∞)

}= fu(6). (5.13)

Proof. If u = nt0 , then from item 3 of Lemma 5.3.5,

min{

fu(t) : t ∈ [6,+∞)

}= fu(tmin),

for some tmin ∈ (t0, t0+1), and fu is decreasing in [6, tmin] and increasing in [tmin,+∞). Therefore,

min{

fu(t) : t ∈ Z∩ [6,+∞)

}= fu(t0) = fu(t0 +1).


Analogously, if u = nt0+1, then

min{

fu(t) : t ∈ Z∩ [6,+∞)

}= fu(t0 +1) = fu(t0 +2).

Thus, in the previous two cases, (5.12) holds.

Now, consider nt0 < u < nt0+1. From item 2 of Lemma 5.3.5, since u < nt0+i, for alli ∈ Z∩ [1,+∞), it is obtained that

fu(t0 +1)< fu(t0 +2)< fu(t0 +3)< · · · .

Also, since nt0−i < u, for all i ∈ {0, . . . , t0−6}, again from item 2 of Lemma 5.3.5, it followsthat

fu(t0 +1)< fu(t0)< fu(t0−1)< fu(t0−2)< · · ·< fu(6).

This completes the proof of (5.12).

Further, if 2 6 u < n6, since n6 < n6+i, for all i ∈ Z∩ [1,+∞), then, from item 2 ofLemma 5.3.5,

fu(6)< fu(7)< fu(8)< · · · .

This proves the equality (5.13). �

Lemma 5.3.7. Let d > 3 be a proper divisor of p−1, and n = (p−1)/d. Then,

min{

fn(t) : t ∈ Z∩ [6,d +3] and t6 n/2+5}= min

{fn(t) : t ∈ Z∩ [6,d +3]

}.

Moreover: if n6 nd+3, then

min{

fn(t) : t ∈ Z∩ [6,d +3] and t6 n/2+5}= min

{fn(t) : t ∈ Z∩ [6,+∞)

}.

Proof. Let t0 = ⌊n/2+5⌋. Then,

t0 =

{n/2+5, if n is even

(n+9)/2, if n is odd

and n < nt0 . Also, nt0 < nt0+i, for all i ∈ Z∩ [1,+∞), implies by item 2 of Lemma 5.3.5 that

fn(t0)< fn(t0 +1)< fn(t0 +2)< fn(t0 +3)< · · · .

Therefore,

min{

fn(t) : t ∈ Z∩ [6,d +3] and t6 n/2+5}= min

{fn(t) : t ∈ Z∩ [6,d +3]

}.

Further, nd+3 < nd+3+i, for all i ∈ Z∩ [1,+∞). In particular, if n6 nd+3, then item 2 ofLemma 5.3.5 implies also that

fn(d +3)6 fn(d +4)< fn(d +5)< fn(d +6)< · · ·

and thus

min{

fn(t) : t ∈ Z∩ [6,d +3] and t6 n/2+5}= min

{fn(t) : t ∈ Z∩ [6,+∞)

},



5.3.1.2 The proof of Theorem 5.3.4

Let d > 3 be a proper divisor of p− 1. If n = (p− 1)/d, from Corollary 5.3.3 and itsproof,

Np(X ) 6 (N−1) · (d2−2d)+d · (p+N)

N−2 · d1 ·ϒ1 +d2 ·ϒ2 +d ·ϒ

N

6 (N−1) · (d2−2d)+d · (p+N)

N− 2d ·ϒ

N

=12s3 + 13

6 s2− 73s−4

s+4·d2 +

4d · (p · (s−1)+1)(s+4) · (s−1)

=12s3 + 13

6 s2− 73s−4

s+4·d2 +

4d · ((p−1) · (s−1)+ s)(s+4) · (s−1)

612s3 + 13

6 s2− 73s−4

s+4·d2 +

4d · ((p−1) · (s−1)+d)(s+4) · (s−1)

=12s3 + 13

6 s2− 73s−4

s+4·d2 +

4d · (d ·n · (s−1)+d)(s+4) · (s−1)

= d2 ·( 1

2s3 + 136 s2− 7

3s−4s+4

+4 ·(

ns+4

+1

(s+4) · (s−1)

))6 d2 ·

( 12s3 + 13

6 s2− 73s−4

s+4+4 · n+1

s+4

),

for each s ∈ {2, . . . ,d−1} satisfying 2d · (s−1)6 p−1, since the latter condition, together withPropositions 3.2.13 and Proposition 5.3.2, implies that DP1,P2

s is Fp-Frobenius classical. Hence,defining t := s+4, it follows that

Np(X )6 d2 ·(

3t2−23t+266

+4 · n+3t

),

for all t ∈ {6, . . . ,d +3} satisfying t6 n2 +5, and

Np(X )

d2 6V (n) := min{

3t2−23t+266

+4 · n+3t : t ∈ {6, . . . ,d +3} and t6 n

2+5}.

Consider the notation given as in Subsection 5.3.1.1. From the hypotheses, n6 nd+3 =(d+3)·(d+4)·(3d−1)

12 −3. Therefore, Lemma 5.3.7 implies the equality

V (n) = min{

fn(t) : t ∈ Z∩ [6,+∞)

}.

The objective is to determine a suitable function W such that W (n)>V (n), for all primenumbers p and for all proper divisors d > 3 of p−1, where n = (p−1)/d. Defining for eachu ∈ [2,+∞)

V (u) := min{

fu(t) : t ∈ Z∩ [6,+∞)

},


Lemma 5.3.6 implies that

V (u) =

{fu(t0 +1), if nt0 6 u6 nt0+1 for some t0 ∈ Z∩ [6,+∞)

fu(6), if 26 u6 n6,

and by Lemma 5.3.5

V (nt0) = fnt0(t0) =

32

t20−

376

t0 +1,

for t0 ∈Z∩ [6,+∞). Thus, one may choose a convenient concave function W such that W (nt0)>

V (nt0) for all t0 ∈ Z∩ [6,+∞), and such that W (u)> V (u), for all integers 26 u < n6. Indeed,fixed u ∈ [n6,+∞), if nt0 6 u6 nt0+1, for some t0 ∈ Z∩ [6,+∞), let u = u1nt0 +u2nt0+1, where06 u1,u2 6 1 and u1 +u2 = 1. Then

W (u) = W (u1nt0 +u2nt0+1)

> u1W (nt0)+u2W (nt0+1) (by the concavity of function W )

> u1V (nt0)+u2V (nt0+1)

= u1fnt0(t0)+u2fnt0+1(t0 +1)

= u1fnt0(t0 +1)+u2fnt0+1(t0 +1)

= u1 ·[

3(t0 +1)2−23(t0 +1)+266

+4 · nt0 +3t0 +1

]+

+ u2 ·[

3(t0 +1)2−23(t0 +1)+266

+4 ·nt0+1 +3

t0 +1

]= (u1 +u2) ·

3(t0 +1)2−23(t0 +1)+266

+4 ·u1nt0 +u2nt0+1 +3 · (u1 +u2)

t0 +1

=3(t0 +1)2−23(t0 +1)+26

6+4 · u+3

t0 +1= fu(t0 +1)

= V (u).

Let us consider the family of concave functions Wλ defined in [2,+∞) by

Wλ (u) = 3 · (21/2 ·u)2/3− 10319· (21/2 ·u)1/3 +λ ,

where λ is a constant.

Direct computation shows that the smallest λ such that Wλ (nt0) > V (nt0) for all t0 ∈Z∩ [6,+∞), and such that Wλ (u)> V (u) = fu(6) for all integers 26 u < n6, is

103− 103

19·21/2 ∼=

133.

Therefore, if W is defined in [2,+∞) by

W (u) = 3 · (21/2 ·u)2/3− 10319· (21/2 ·u)1/3 +

133

then the proof follows. �


5.4 Affinely regular polygons

5.4.1 A brief introduction to affinely regular polygons

This subsection is based on the survey article (KORCHMÁROS; SZONYI, 2008) and insome references therein. Here, F denotes an arbitrary field of characteristic p> 0 and A2(F) isthe affine plane over F.

Definition 5.4.1. An n-gon in A2(F) is a set of n pairwise distinct points arranged in a cyclicorder. If no three vertices of it are collinear, then the n-gon is called nondegenerate.

From now on, only nondegenerate n-gons are considered.

Definition 5.4.2. If B = B1B2 . . .Bn is a regular n-gon in the Euclidean plane, then an n-gonA = A1A2 . . .An in the affine plane A2(F) is affinely regular if the bijection Bi ↦→ Ai preserves allparallelisms between chords (sides and diagonals). Symbolically,

Bi0Bk0 ‖ Bi1Bk1 ⇔ Ai0Ak0 ‖ Ai1Ak1,

for all 16 i0 < k0 6 n and 16 i1 < k1 6 n.

The first part of the following theorem presents a remarkable property pointed outin (KORCHMÁROS, 1974) and independently in (CRAATS; SIMONIS, 1986). Further, theclassification in the second part can be found in (KORCHMÁROS, 1976a) and (KORCHMÁROS,1976b).

Theorem 5.4.3. (KORCHMÁROS; SZONYI, 2008, Theorem 2.1 and Lemma 2.2). Everyaffinely regular polygon in A2(F) is inscribed in an irreducible conic. More specifically, theaffinely regular polygons in A2(F) are exactly given by the following examples and their imagesunder the affinities of A2(F).

1. (Hyperbola). Let H : XY = 1 and consider G a finite multiplicative subgroup of F of ordern. If α ∈ F* and g is a generator of G, then A = A1A2 . . .An is an affinely regular n-goninscribed in H, where Ai = (giα,(giα)−1) for each i ∈ {1,2, . . . ,n}.

2. (Parabola, p > 0). Let P : Y = X2 and consider α ∈ F. Then A = A1A2 . . .Ap is an affinelyregular p-gon inscribed in P, where Ai = (α + i,(α + i)2) for each i ∈ {1,2, . . . ,p}.

3. (Ellipse). Let T 2− γT + 1 be an irreducible polynomial over a field F and considerE : X2 +Y 2− γXY = 1. If θ is such that θ 2− γθ + 1 = 0 and E = F(θ) is a quadraticextension of F, then, identifying (α,β ) ∈ A2(F) with α +βθ ∈ E, the points of E arethose elements α +βθ ∈ E which satisfy the equation (α +βθ) · (α +βθ−1) = 1. Now,considering α + βθ ∈ E* and g a generator of a multiplicative subgroup G of E oforder n, it follows that A = A1A2 . . .An is an affinely regular n-gon inscribed in E, whereAi = gi(α +βθ).

5.4. Affinely regular polygons 117

The following result was first proved in (KORCHMÁROS, 1974).

Theorem 5.4.4. (KORCHMÁROS; SZONYI, 2008, Theorem 2.3). In A2(Fq), an affinelyregular n-gon exists if and only if one of the following cases occurs:

1. n | (q−1) and the n-gon is inscribed in a hyperbola.

2. n = p and the n-gon is inscribed in a parabola.

3. n | (q+1) and the n-gon is inscribed in an ellipse.

Now, from (KORCHMÁROS, 1983), (SZONYI, 1987) and (KORCHMÁROS; STORME;SZONYI, 1997), the subsequent result follows.

Theorem 5.4.5. (KORCHMÁROS; SZONYI, 2008, Theorem 5.9). Let A = A1A2 . . .An be anaffinely regular n-gon in A2(Fq) inscribed in the irreducible conic C. If q/n < q1/4/2, then thechords of A cover all points of A2(Fq) except the points of C and possibly the center of C.

The previous result raises a natural question: what is the number of chords of an affinelyregular n-gon of A2(Fq) passing through a given point distinct from its vertices?

In the particular case of a n-gon A = A1A2 . . .An inscribed in a hyperbola (resp. anellipse), it was shown that determining the number of chords of A passing through a point distinctfrom its vertices is equivalent to determine the number of rational affine points lying in anappropriate subset of the plane (resp. the number of rational affine points) of a curve of the formof curve F . More precisely, from (ABATANGELO; KORCHMÁROS, 1997, Proposition 3.2)and (GIULIETTI, 2008, Proposition 2.1), the following result holds.

Theorem 5.4.6. Let A = A1A2 . . .An be an affinely regular n-gon in A2(Fq).

1. Suppose that A is inscribed in a hyperbola H. Up to a change of coordinates, it is possibleto assume that H has equation XY = 1, and that Ai = (γ i,γ−i) for each i ∈ {1,2, . . . ,n},where γ is a primitive n-th root of unity in Fq. If (α,β ) ∈ A2(Fq)∖H and n(α,β ) denotesthe number of chords of A passing through (α,β ), then

n(α,β ) =N

2d2 ,

where d = (q− 1)/n and N is the number of Fq-rational affine points not lying on thecoordinate axes or the line of equation X = Y of the geometrically irreducible plane curvegiven in affine coordinates by

αXdY d−Xd−Y d +β = 0.

2. Suppose that A is inscribed in an ellipse E. Up to a change of coordinates, one mayassume that E has equation X2 +Y 2− γXY = 1, where γ ∈ Fq is such that T 2− γT + 1is irreducible over Fq. Further, identifying A2(Fq) with Fq2 as in Theorem 5.4.3, after a


change of coordinates, it is possible to assume that A1,A2, . . . ,An are the d-th powers inFq2 , with d = (q2−1)/n. If (α,β ) ∈ A2(Fq) and n(α,β ) denotes the number of chords ofA passing through (α,β ), then ∣∣∣∣n(α,β )−

N2d2

∣∣∣∣6 2,

where N is the number of Fq2-rational affine points of the geometrically irreducible planecurve given in affine coordinates by

(α +βθ)qXdY d−Xd−Y d +(α +βθ) = 0,

and θ is a root of T 2− γT +1 in Fq2 .

Based on the previous result and using the Stöhr-Voloch theory, in (ABATANGELO;KORCHMÁROS, 1997) and (GIULIETTI, 2008) bounds for the number of chords of affinelyregular polygons inscribed in a hyperbola or an ellipse were studied. In the next subsection,for the particular case of an affinely regular n-gon A = A1A2 . . .An in A2(Fp) inscribed in ahyperbola, an upper bound for the number n(a,b) of chords of A passing through (a,b) ∈ A2(Fp)

distinct from its vertices is provided.

5.4.2 Number of chords of an affinely regular polygon inscribed in ahyperbola passing through a given point

Let Fq be the prime field Fp and consider A = A1A2 . . .An an affinely regular polygonwith n = (p−1)/d vertices inscribed in a hyperbola H of A2(Fp). As presented in Subsection5.4.1, changing variable, one may consider H given by the equation XY = 1.

From Theorems 5.3.4 and 5.4.6, the following upper bound for the number n(a,b) ofchords of A passing through (a,b) ∈ A2(Fp) distinct from its vertices is obtained.

Corollary 5.4.7. If d > 3 is a proper divisor of p−1 such that

p−16 d ·((d +3) · (d +4) · (3d−1)

12−3),

then

n(a,b) 612·(

3 · (21/2 ·n)2/3− 10319· (21/2 ·n)1/3 +

133

). (5.14)

Remark 5.4.8. One can check that bound (5.14) improves that given in (5.5) for n > 44. Onthe other hand, for each 26 n < 44, bounds (5.5) and (5.14) effectively differ by at most oneunit. Moreover, considering the definition of V (n) = V (n) as given in the proof of Theorem5.3.4, for 25 < n < 44, it is obtained that n(a,b) 6 ⌊1

2V (n)⌋ 6 ⌊n+13 ⌋, with n+1

3 as in (5.5). Foran illustration of the region where (5.14) is better than the bound for n(a,b) derived from theHasse-Weil bound, see also Figure 1.

5.4. Affinely regular polygons 119

Figure 1 – Contour plot of the difference ∆(p,d) = ( p+1+2·(d−1)2·p1/2

2·d2 )− 12 · (3 · (2

1/2 ·n)2/3− 10319 · (2

1/2 ·n)1/3 + 13

3 ) between the bounds for the number n(a,b) obtained from the Hasse-Weil bound

and Corollary 5.4.7, for d < p−16 d · ( (d+3)·(d+4)·(3d−1)12 −3) and n = (p−1)/d. In red, the

curve p−1 = d · ( (d+3)·(d+4)·(3d−1)12 −3).

121

CHAPTER

6ON THE ZETA FUNCTION AND THE

AUTOMORPHISM GROUP OF THEGENERALIZED SUZUKI CURVE

The present chapter is an extended and adapted version of (BORGES; COUTINHO,Preprint 2019).

6.1 IntroductionFor a nonsingular geometrically irreducible curve Y of genus g defined over the finite

field Fq, where q is a power of a prime number p, consider the problem of determining thenumber Nq(Y ) of Fq-rational points on Y . In this direction, the remarkable Hasse-Weil bound(see Section 2.7) states that

|Nq(Y )− (q+1)|6 2gq1/2.

Curves attaining the previous upper (resp. lower) bound are called Fq-maximal (resp.Fq-minimal). For p = 2, an important example is the Deligne-Lusztig curve associated to theSuzuki group Sz(q) (see (DELIGNE; LUSZTIG, 1976), (HANSEN; STICHTENOTH, 1990),(HANSEN, 1992)), here for simplicity called the Suzuki curve, which is the nonsingular modelYS defined over Fq of the plane curve1

S : Y q−Y = Xq0(Xq−X), (6.1)

where q0 = 2t and q = 22t−1. Indeed, in (HANSEN, 1992, Proposition 4.3), together with theexpression for the Zeta function of YS , the explicit formula for the number of rational points onYS shows that it is Fq4-maximal.1 This plane model for the Suzuki curve is not the most commonly found in the literature, where in

general q0 = pt and q = p2t+1. However, from (GARCIA; STICHTENOTH, 1991, Proposition 4.2),these two representations are equivalent.

122 Chapter 6. On the Zeta function and the automorphism group of the generalized Suzuki curve

The Suzuki curve is optimal in the sense that its number of Fq-rational points coin-cides with the maximum number of Fq-rational points that a curve of its genus can have (see(HANSEN; STICHTENOTH, 1990, Proposition 2.1)). Moreover, in (FUHRMANN; TORRES,1998, Theorem 5.1), it is shown that this curve can be characterized by its genus and number ofFq-rational points.

In addition to its maximality and optimality properties, the Suzuki curve is also knownfor its large automorphism group with respect to the genus. Specifically, it is one of the fourexamples of curves of genus g> 2 having an automorphism group of size greater than or equal to8g3 (see (HENN, 1978), (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 11.127)).Furthermore, in (GIULIETTI; KORCHMÁROS; TORRES, 2006) and (BASSA et al., 2013), theinvestigation of genera and plane models for quotients of the Suzuki curve is addressed. Otherstudies examining embeddings in PN (see (BALLICO; RAVAGNANI, 2015), (DUURSMA;EID, 2015)), class field theory (LAUTER, 1999), the invariant a-number (FRIEDLANDER et

al., 2013), the Weierstrass semigroups and coding theory (see (KIRFEL; PELLIKAAN, 1995),(CHEN; DUURSMA, 2003), (MATTHEWS, 2004), (GIULIETTI; KORCHMÁROS, 2008),(DUURSMA; PARK, 2012), (EID et al., 2016), (BARTOLI; MONTANUCCI; ZINI, 2018))were also carried out over the past years. More recently, the construction in (SKABELUND,2018) of a Galois cover of the Suzuki curve raised a number of other issues to be investigated(see (GIULIETTI et al., 2018), (MONTANUCCI; TIMPANELLA; ZINI, 2018)).

Most of the aforementioned applications was based on the knowledge of the numberof Fqn-rational points and the automorphism group of the Suzuki curve. Motivated by this, letp > 2, q0 = pt and q = pm, where m, t are positive integers satisfying the relation m = 2t−1. IfGS is the projective geometrically irreducible (see Proposition 6.3.12) plane curve defined overFq given in affine coordinates by

GS : Y q−Y = Xq0(Xq−X), (6.2)

let XGSbe its nonsingular model defined over Fq, which is called here the generalized Suzuki

curve.

The objective of this chapter is twofold. On the one hand, the number of Fqn-rationalpoints on XGS

is investigated2. As a consequence, the L-polynomial over Fq of XGSis recovered.

The L-polynomial of a curve Y defined over Fq encodes information on the order of the groupPic0(Fq(Y )), the zero-degree Fq-divisor class group. Using this fact, some constructions ofcurves with many rational points are presented in (VOLOCH, 2000). Moreover, considering theFq-Frobenius endomorphism Φ on the Jacobian variety of Y , then the characteristic polynomial

2 We point out here that for n = 1, the number Nqn(XGS) was already studied in (PEDERSEN;

S∅RENSEN, 1990) in connection with geometric Goppa codes. Also, for p = 3 and considering(GARCIA; STICHTENOTH, 1991, Proposition 4.2), the curve XGS

is Fq-covered by the so-calledRee curve, and then the information on its maximality given by Theorem 6.1.1 could be recovered byTheorem 2.7.4.

6.1. Introduction 123

of Φ is described exactly by the reciprocal of the L-polynomial of Y over Fq, and from itsfactorization the degree of the Frobenius linear series on Y is obtained. For further details onthis topic, see (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Sections 9.7 and 9.8).

On the other hand, the automorphism group of XGSis presented. More generally, for p

an arbitrary prime number, q = pm and q0 = pt , where m, t are positive integers satisfying therelation 2t > m> t, with m = 2t−1 if p = 2, we note that considering X the nonsingular modeldefined over Fq of

Y q−Y = Xq0(Xq−X),

a straightforward verification shows that our main result on the automorphism group of XGS,

Theorem 6.1.3 below, also holds for X . Consequently, for p = 2 this extension of Theorem6.1.3 completes the description of the whole automorphism group of X started in (GIULIETTI;KORCHMÁROS, 2014).

The following are the main results of this work.

Theorem 6.1.1. If g denotes the genus of XGS, then the number Nqn(XGS

) of Fqn-rationalpoints on XGS

is described as follows.

1. If p | n, then

Nqn(XGS) =

qn +1, if n is oddqn +1−2gqn/2, if n is even and p≡ 1 (mod 4)qn +1−2gqn/2, if n≡ 0 (mod 4) and p≡ 3 (mod 4)qn +1+2gqn/2, if n≡ 2 (mod 4) and p≡ 3 (mod 4).

2. If p - n, then

Nqn(XGS) =

qn +1, if n is even

qn +1+2gqn/2 p−1/2((−1)(n−1)/2n

p

), if n is odd

,

where(*p

)is the Legendre symbol.

Now, define

p1/2 :=

{p1/2, if p≡ 1 (mod 4)ip1/2, if p≡ 3 (mod 4)

. (6.3)

Theorem 6.1.2. If g denotes the genus of XGSand Mp(T ) is the minimal polynomial of

−ζp/p1/2 over Q, where ζp is the primitive p-th root of unity e2πip , then the L-polynomial

LXGS(T ) of XGS

over Fq is given by

LXGS(T ) =

(pp−1((−1)

p+12 +qT 2) ·Mp(pt−1T )2

) gp

.


Finally, if Fq(XGS) is the function field of XGS

and x,y ∈ Fq(XGS) are such that

Fq(XGS) = Fq(x,y) and yq−y = xq0(xq−x), then the automorphism group of XGS

is describedas follows.

Theorem 6.1.3. The automorphism group G of XGShas order q2(q−1) and is given by the

maps

(x,y) ↦→ (αx+β ,αβq0x+α

q0+1y+ γ),

where α ∈ F*q and β ,γ ∈ Fq. Moreover,

G=G1 oH,

where G1 is the Sylow p-subgroup of G consisting of the maps

(x,y) ↦→ (x+β ,β q0x+ y+ γ),

with β ,γ ∈ Fq, and H is the cyclic complement of G1 in G, which can be described by the maps

(x,y) ↦→ (αx,αq0+1y),

with α ∈ F*q.

This chapter is organized as follows. In Section 6.3, the general background necessary toprove Theorems 6.1.1, 6.1.2 and 6.1.3 is established. The proofs are presented in Sections 6.4,6.5 and 6.7, respectively. Further, in Section 6.6 some examples of the L-polynomial of XGS

for p = 3,5,7,11,13,17 and 19 are considered, and an application of this fact is presented inSection 6.8.

6.2 NotationThe following notation is used throughout this text.

∙ p is a prime number, q = pm and q0 = pt for some positive integers m, t.

∙ For each positive integer n, Fqn is the finite field with qn elements and F*qn := Fqn ∖{0}.

∙ TrFqn/Fp and NFqn/Fp denote the absolute trace and norm functions of Fqn , respectively.

∙ The word curve means a projective nonsingular geometrically irreducible algebraic curve.Also, as established in Notation 1.1.3, the term plane curve means a projective planealgebraic curve.

∙ For a curve Y , Nqn(Y ) denotes its number of Fqn-rational points.

∙ e is the Euler’s number, i is the imaginary unity and ζk := e2πik for each positive integer k.

∙ For each positive integer k,

µk =

{ζ

ik : 16 i6 k and gcd(i,k) = 1

}is the set of roots of the k-th cyclotomic polynomial.

6.3. Preliminaries 125

6.3 Preliminaries

6.3.1 L-polynomials and supersingular curves

Let Y be a curve of genus g defined over the finite field Fq, and let LY (T ) be theL-polynomial (defined over Fq) of Y , which is the numerator of the Zeta function (over Fq) ofY (see Section 2.6).

If ω1, . . . ,ω2g ∈ C are the roots of the reciprocal polynomial of LY (T ), then fromTheorem 2.6.13 the following holds:

Nqn(Y ) = qn +1−2g

∑i=1

ωni . (6.4)

Recall that from the Hasse-Weil theorem (see Theorem 2.7.1), |ωi| = q1/2, for all i =

1, . . . ,2g, and then ωi = q1/2ξi, with |ξi|= 1 for each i. In particular, (6.4) can be rewritten as

Nqn(Y ) = qn +1−qn/22g

∑i=1

ξni , (6.5)

and Y is Fq2n-maximal (resp. Fq2n-minimal) if and only if ξ 2ni = −1 (resp. ξ 2n

i = 1) for eachi ∈ {1, . . . ,2g}.

Definition 6.3.1. The curve Y is supersingular if ξi, with i = 1, . . . ,2g, are roots of unity. Inthis case, the number

s := min{

n : ξni = 1 for all i = 1, . . . ,2g

}is called the period of Y over Fq.

This subsection ends with the following result, which is important to prove Theorem6.1.1.

Theorem 6.3.2. (MCGUIRE; YILMAZ, 2018, Theorem 1). Let Y be a supersingular curve ofgenus g defined over Fq with period s. Let n be a positive integer, let n1 = gcd(n,s) and writen = n1k. If q is odd, then the following occurs:

1. If n1m is even, then

Nqn(Y )− (qn +1) = q(n−n1)/2[

Nqn1 (Y )− (qn1 +1)].

2. If n1m is odd and p | k, then

Nqn(Y )− (qn +1) = q(n−n1)/2[

Nqn1 (Y )− (qn1 +1)].


3. If n1m is odd and p - k, then

Nqn(Y )− (qn +1) = q(n−n1)/2[

Nqn1 (Y )− (qn1 +1)](

(−1)(k−1)/2kp

),

where(*p


6.3.2 Elementary abelian p-extensions of algebraic fuction fields

Let Σ2/Σ1 be a Galois extension of algebraic function fields and suppose that Σ1 hascharacteristic p > 0.

Definition 6.3.3. The extension Σ2/Σ1 is an elementary abelian p-extension if its Galois group,Gal(Σ2/Σ1), is elementary abelian of exponent p. In this case, the degree of the extension,denoted here by [Σ2 : Σ1], is equal to q = pm for some positive integer m.

The following result provides a description of elementary abelian p-extensions Σ2/Σ1 ofdegree q = pm in cases in which Σ1 contains the finite field Fq.

Proposition 6.3.4. (GARCIA; STICHTENOTH, 1991, Propositions 1.1 and 1.2). Suppose thatΣ2/Σ1 is an elementary abelian p-extension of degree q and that Fq ⊆ Σ1. Then, there is anelement y ∈ Σ2 such that Σ2 = Σ1(y) and whose minimal polynomial over Σ1 is of the form

T q−T − z

for some z ∈ Σ1. Conversely, if Fq ⊆ Σ1 and, for some z ∈ Σ1, the polynomial

T q−T − z

is irreducible over Σ1, then the extension Σ1(y)/Σ1, with y satisfying yq− y− z = 0, is anelementary abelian p-extension of degree q. Further, the intermediate fields Σ1 ⊆ Σ⊆ Σ1(y) with[Σ : Σ1] = p are the following ones: Σ = Σµ := Σ1(yµ), where µ ∈ F*q,

yµ := (µy)pm−1+(µy)pm−2

+ · · ·+(µy)p +µy,

and yµ satisfies the equation

ypµ −yµ = µz.

Remark 6.3.5. Considering the notation as in Proposition 6.3.4, we have that Σµ1 = Σµ2 if andonly if µ1 = µµ2 for some µ ∈ F*p. Therefore, there are exactly

q−1p−1

intermediate fields Σ of Σ1(y)/Σ1 such that [Σ : Σ1] = p.


Now, let Σ2/Σ1 be an elementary abelian p-extension of degree q and suppose that Σ1

contains the finite field Fq. Also, let{Σµi : i = 1, . . . ,

q−1p−1

}be the set of all intermediate fields of Σ2/Σ1, and let Y2 and Yµi , with i = 1, . . . , q−1

p−1 , be thenonsingular models defined over Fq of Σ2 and Σµi , respectively.

This subsection ends with the following result, which relates the L-polynomial (overFq) of Y2 with the L-polynomials (over Fq) of Yµi , for i = 1, . . . , q−1

p−1 . As a consequence, onecan express the number of Fqn-rational points on Y2 as a function of the number of Fqn-rationalpoints on Yµi , where i = 1, . . . , q−1

p−1 , for each positive integer n.

Proposition 6.3.6. (DUURSMA; STICHTENOTH; VOSS, 1996, Corollary 6.7) Considering

the previous notation, then it follows that LY2(T ) =

q−1p−1

∏i=1

LYµi(T ). In particular,

Nqn(Y2)− (qn +1) =

q−1p−1

∑i=1

(Nqn(Yµi)− (qn +1)

)for each positive integer n.

6.3.3 On the number of Fqn-rational points of Y p−Y = XR(X)

Suppose that p is an odd prime number. For each positive integer n, the number ofFqn-rational points on the nonsingular model YR defined over Fq of the plane curve given inaffine coordinates by

FR : Y p−Y = XR(X),

where R(X) is a p-polynomial defined over Fq, that is,

R(X) = αkX pk+αk−1X pk−1

+ · · ·+α0X ,

with αk,αk−1, . . . ,α0 ∈ Fq and αk = 0, is studied.

The following proposition collects some properties of FR.

Proposition 6.3.7. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Lemma 12.1). Theplane curve FR is geometrically irreducible. Moreover:

1. The genus g of FR is given by

g =pk(p−1)

2.


2. P = (0 : 1 : 0) ∈ P2(Fq) is the unique point at infinity of FR. If k = 1, then FR is nonsin-gular. Otherwise, P is the unique singular point of FR. In both cases, P has multiplicitypk− p+1 and it is the center of only one Fq-rational branch of FR.

Corollary 6.3.8. The number of Fqn-rational points on YR is given by the number of solutionsin F2

qn of the equation

Y p−Y = XR(X) (6.6)

added to 1.

The main reference from now until the end of this subsection is (BOUW et al., 2016,Section 2). To count the number of solutions in F2

qn of (6.6), set

B(n)R : F2

qn → Fp

(α,β ) ↦→ 12TrFqn/Fp(αR(β )+βR(α))

.

One can check that B(n)R is an Fp-symmetric bilinear form in Fqn , with associated quadratic

form

Q(n)R : Fqn → Fp

α ↦→ TrFqn/Fp(αR(α)),

which satisfies Q(n)R (λα) = λ 2Q(n)

R (α) for all λ ∈ Fp, and also

B(n)R (α,β ) =

12

Q(n)R (α +β )− 1

2Q(n)

R (α)− 12

Q(n)R (β )

for each (α,β ) ∈ F2qn .

Considering the radical (kernel) of B(n)R

W (n)R =

{α ∈ Fqn : B(n)

R (α,β ) = 0 for each β ∈ Fqn

},

which is an Fp-linear subspace of Fqn , the following properties hold.

Proposition 6.3.9. (BOUW et al., 2016, Proposition 2.1). The following occurs.

1. Let α ∈ Fqn . If α ∈W (n)R , then Q(n)

R (α) = 0.

2. The set W (n)R consists of the roots in Fqn of the polynomial

ER(T ) = R(T )pk+

k

∑i=0

(αiT )pk−i.

Based on Corollary 6.3.8, a connection between the number of Fqn-rational points on YR

and the dimension of V (n)R := Fqn/W (n)

R over Fp is now established.


Proposition 6.3.10. (BOUW et al., 2016, Proposition 2.4). Let B(n)R be the projection of B(n)

R

onto V (n)R , that is, define

B(n)R (α +W (n)

R ,β +W (n)R ) := B(n)

R (α,β )

for each α +W (n)R ,β +W (n)

R ∈V (n)R . Then, B(n)

R is a nondegenerate bilinear form on V (n)R ×V (n)

R .

As a consequence of the previous result, it follows that the quadratic form

Q(n)R : V (n)

R → Fp

α +W (n)R ↦→ TrFqn/Fp(αR(α))

,

is nondegenerate, and therefore its zero locus{α +W (n)

R ∈V (n)R : Q(n)

R (α) = 0},

defines a smooth quadric over Fp. In (JOLY, 1973), the author provides an expression for thecardinality of a nondegenerate quadratic form. Using this, then the number of solutions in F2

qn of

(6.6) in terms of the dimension of V (n)R over Fp,

dimFp(V(n)R ) = dimFp(Fqn)−dimFp(W

(n)R ) = mn−dimFp(W

(n)R ),

can be described by the following proposition.

Proposition 6.3.11. (BOUW et al., 2016, Proposition 2.6). The number of solutions in F2qn of

(6.6) is

1. qn, if mn−dimFp(W(n)R ) is odd.

2. qn± (p−1)pdimFp(W(n)R )/2qn/2, if mn−dimFp(W

(n)R ) is even.

6.3.4 The curve GS

Let GS be given as in (6.2). From (PEDERSEN; S∅RENSEN, 1990, Theorem 1),(GARCIA; STICHTENOTH, 1991, Proposition 4.1) and a straightforward calculation, thefollowing result holds.

Proposition 6.3.12. GS is a geometrically irreducible plane curve of genus g given by

g =q0(q−1)

2.

Further, P = (0 : 1 : 0) ∈ P2(Fq) is the unique point at infinity of GS , which is also its onlysingular point, having multiplicity equal to q0.

Consider XGSthe nonsingular model of GS defined over Fq, and let Fq(XGS

) be itsfunction field. If x,y ∈ Fq(XGS

) are such that Fq(XGS) = Fq(x,y) and yq− y = xq0(xq− x),

then the following occurs.


Proposition 6.3.13. (DEOLALIKAR, 2002, Theorem 3.3). The extension Fq(x,y)/Fq(x) is aGalois extension of degree q. Moreover, the only ramified place of Fq(x) is the infinite placeP∞, which is totally ramified in Fq(x,y).

Using the one-to-one correspondence between (Fq-rational) points on XGSand (Fq-

rational) places of Fq(XGS) (see Proposition 2.5.9), let Q∞ be the Fq-rational point (see Subsec-

tion 2.3.1) of XGScorresponding to the unique (Fq-rational) place Q∞ of Fq(x,y) lying over the

infinity place P∞ of Fq(x). Then, the following holds.

Proposition 6.3.14. (PEDERSEN; S∅RENSEN, 1990, Proof of Lemma 2). The pole divisorsof x and y are given by

div(x)∞ = qQ∞ and div(y)∞ = (q+q0)Q∞,

respectively.

This subsection ends with the following two results.

Theorem 6.3.15. (PEDERSEN; S∅RENSEN, 1990, Theorem 7). The sets {1,x} and {1,x,y}are bases for the Riemann-Roch spaces L (qQ∞) and L ((q+q0)Q∞), respectively.

Corollary 6.3.16. P = (0 : 1 : 0) is the center of a unique Fq-rational branch of GS , namely thebranch of GS corresponding to the point Q∞ ∈XGS

.

6.3.5 Automorphism group

Let Y be a curve of genus g defined over the finite field Fq.

Definition 6.3.17. The automorphism group G of Y is defined as the group of Fq-automorphismsof the function field Fq(Y ).

Definition 6.3.18. Considering the action of G on the points of Y (see (HIRSCHFELD; KO-RCHMÁROS; TORRES, 2008, Section 11.1)) for Q ∈ Y , the stabilizer of G at Q, denoted byGQ, is the subgroup of G consisting of all automorphisms fixing Q under this action. Further, foreach non-negative integer i, the i-th ramification group G(i)

Q at Q is defined by

G(i)Q =

{σ ∈GQ : vQ(σ(t)− t)> i+1

},

where t ∈ Fq(Y ) is a local parameter at Q and G(i)Q ⊇G(i+1)

Q for all i> 0.

The following result summarizes properties of these subgroups.

Theorem 6.3.19. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Lemma 11.44 and Theo-rem 11.74). The equality G(0)

Q =GQ holds. Moreover, G(1)Q is the unique Sylow p-subgroup of

GQ and GQ =G(1)Q oH, where H is a cyclic subgroup of GQ of order coprime to p.

6.4. Proof of Theorem 6.1.1 131

Lastly, the following result is used in the proof of Theorem 6.1.3.

Theorem 6.3.20. (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem 11.140). Sup-pose that g> 2, and let Q ∈ Y be an Fq-rational point satisfying

|G(1)Q |> 2g+1.

Then, one of the following cases occurs:

1. G=GQ.

2. Y is birationally equivalent to one of the following curves:

a) the Hermitian curve over Fq

H : Y q−Y = Xq+1 (6.7)

b) the Suzuki curve over Fq, given by the nonsingular model defined over Fq of (6.1)c) the Ree curve over Fq, given by the nonsingular model defined over Fq of

R : Y q2− [1+(Xq−X)q−1]Y q +(Xq−X)q−1Y −Xq(Xq−X)q+3q0 = 0, (6.8)

where q = 3m, q0 = 3t and m = 2t−1.

6.4 Proof of Theorem 6.1.1

Using the notation introduced in Subsection 6.3.4, the proof of Theorem 6.1.1 is presented.To better organize the ideas, it is separated in a sequence of steps.

Step 1: elementary abelian p-extension

From Proposition 6.3.4, Fq(x,y)/Fq(x) is an elementary abelian p-extension of degree q.The set of intermediate fields Fq(x)⊆ E ⊆ Fq(x,y) such that [E : Fq(x)] = p is described by{

Eα : α ∈ F*q},

where for each α ∈ F*q, Eα is the intermediate field of Fq(x,y)/Fq(x) given by

Eα := Fq(x,yα),

and yα := (αy)pm−1+(αy)pm−2

+ · · ·+(αy)p+αy satisfies the equation ypα−yα = αxq0(xq−x).

Further, from Remark 6.3.5, let{

αi : i = 1, . . . ,q−1p−1

}⊆ F*q be such that

{Eα : α ∈ F*q

}=

{Eαi : i = 1, . . . ,

q−1p−1

}.


For each α ∈ F*q, consider the plane model (Fα ,(x,yα)) defined over Fq of Eα , whereFα is the geometrically irreducible3 plane curve defined over Fq given in affine coordinates by

Fα : Y p−Y = αXq0(Xq−X).

Also, let Xα be the nonsingular model of Fα defined over Fq, and let LXα(T ) be the L-

polynomial of Xα over Fq.

From Proposition 6.3.6, the following occurs.

Corollary 6.4.1. For each positive integer n, Nqn(XGS)− (qn +1) =

q−1p−1

∑i=1

(Nqn(Xαi)− (qn +1)

).

The following lemmas are important to describe the number Nqn(XGS) for each positive

integer n.

Lemma 6.4.2. For each α ∈ F*q, Eα is Fq-isomorphic to E1, with the Fq-isomorphism inducingan Fq-isomorphism between Fq(x,yα) and Fq(x,y1). In particular, from Corollary 6.4.1,

Nqn(XGS)− (qn +1) =

q−1p−1

(Nqn(X1)− (qn +1)


Proof. Recall that q = pm and q0 = pt , where m = 2t−1, which implies that m− t = t−1. Forα ∈ F*q, let β := α pm−t+pm−t−1+···+p+1 ∈ Fq and γ := NFq/Fp(α) ∈ Fp. Then

βq0β =

(α

pm−t+pm−t−1+···+p+1)pt

αpm−t+pm−t−1+···+p+1

= αpm+pm−1+···+pt+1+pt

·α pm−t+pm−t−1+···+p+1

= αpm+pm−1+···+pt+1+pt

·α pt−1+pt−2+···+p+1

= αpm

αpm−1+···+p+1

= αγ,

Eα = Fq(x,yα) = Fq(βx,γyα) and Fq(x,yα) = Fq(βx,γyα), with

(βx)q0((βx)q− (βx)) = βq0βxq0(xq− x)

= αγxq0(xq− x)

= γ(ypα − yα)

= (γyα)p− (γyα).

Therefore,

Eα → E1A(βx,γyα )B(βx,γyα )

↦→ A(x,y1)B(x,y1)

,

3 The geometric irreducibility of Fα for each α ∈ F*q follows from the geometric irreducibility of GS

(GARCIA; STICHTENOTH, 1991, Lemma 1.3).


where A(X ,Y ),B(X ,Y )∈Fq[X ,Y ] and B(βx,γyα) = 0, is an Fq-isomorphism between Eα and E1

that induces an Fq-isomorphism between Fq(x,yα) and Fq(x,y1). This completes the proof. �

Lemma 6.4.3. For each α ∈ F*q, E1 = Fq(x,z1) and Fq(x,y1) = Fq(x,z1), where z1 ∈ Fq(x,y1)

is defined by

z1 := y1− xqp x

q0p − x

qp2 x

q0p2 −·· ·− x

qq0 x

q0q0

and satisfies zp1 − z1 = x(x

qq0 − xq0). In particular, if F1 is the geometrically irreducible plane

curve given by

F1 : Y p−Y = X(Xq

q0 −Xq0)

and X1 is its nonsingular model defined over Fq, then (F1,(x,z1)) is another plane modeldefined over Fq of E1 and

Nqn(XGS)− (qn +1) =

q−1p−1

(Nqn(X1)− (qn +1)


Proof. The equalities E1 = Fq(x,z1) and Fq(x,y1) = Fq(x,z1) follow immediately from thedefinition of z1. Also, we have that

zp1 − z1 = yp

1 − y1− xqxq0 + xq

q0 xq0q0

= xq0(xq− x)− xqxq0 + xq

q0 x

= x(xq

q0 − xq0),

as desired. Finally, the other statements follow from the geometric irreducibility of F1 given byProposition 6.3.7. �

Step 2: the number of Fqn-rational points on F1 : Y p−Y = X(Xq

q0 −Xq0)

Considering the notation given as in Subsection 6.3.3, let F1 = FR, where R(X) =

Xq

q0 −Xq0 .

To apply Proposition 6.3.11, let us determine the dimension of W (n)R over Fp based on

the characterization provided by Proposition 6.3.9. For this, first note that here

ER(T ) = (T −T q)p− (T −T q).

and thus, the following statements are equivalent for an element α ∈ Fqn :

1. α ∈W (n)R ;

2. ER(α) = 0;

3. αq−α ∈ Fp.


This part starts with the following lemma, which is a consequence of the results in(COULTER; HENDERSON, 2004) and (LIANG, 1978).

Lemma 6.4.4. Let β ∈ Fp be fixed. Then

#{

α ∈ Fqn : αq−α−β = 0

}=

{q, if nβ = 00, otherwise.

In particular, the splitting field of ER(T ) is Fqp ,

#W (n)R =

{pq, if p | nq, otherwise

,

and

dimFp(W(n)R ) =

{m+1, if p | nm, otherwise.

Recall that m is odd. Therefore, since Proposition 6.3.7 provides that the genus of X1 isgiven by

g(X1) := pt(p−1)/2,

as a consequence of Corollary 6.3.8, Proposition 6.3.11, and Lemma 6.4.4, the following holds.

Lemma 6.4.5. The number of Fqn-rational points on X1 can be described as follows.

1. If p | n, then

Nqn(X1) =

{qn +1, if n is oddqn +1±2g(X1)qn/2, if n is even.

(6.9)

2. If p - n, then

Nqn(X1) =

{qn +1, if n is evenqn +1±2g(X1)qn/2 p−1/2, if n is odd.

(6.10)

In particular, X1 is Fqn-maximal or minimal if and only if p | n and n is even, and also Nq(X1) =

qp+1.

Therefore, to determine the sign ± in (6.9) and (6.10) it is necessary to separate theanalysis in two cases.

The case p | n

Since the splitting field of ER(T ) is the field Fqp , with the following result one candetermine the sign in (6.9).


Lemma 6.4.6. (BOUW et al., 2016, Comments on Page 105 and Theorem 7.4). There existδ ∈ Fqp and an Fqp-rational morphism φ : X1 →Xδ , where Xδ is the nonsingular modeldefined over Fqp of

Y p−Y = δX2.

Based on Lemma 6.4.6, consider all the structures X1, Xδ and φ defined over theextensions Fqn of Fqp , with n even. In particular, δ is a square in all these extensions, and sincem is odd, the following occurs.

Proposition 6.4.7. (BOUW et al., 2016, Lemma 9.1). Let n be an even positive integer suchthat p | n. Then, Xδ is Fqn-maximal if and only if

p≡ 3 (mod 4) and n≡ 2 (mod 4).

Therefore, from Theorem 2.7.4, the sign in (6.9) is obtained.

Lemma 6.4.8. Let n be an even positive integer such that p | n. The curve X1 is Fqn-maximal ifand only if p≡ 3 (mod 4) and n≡ 2 (mod 4). In particular, (6.9) can be rewritten in the form

Nqn(X1) =

qn +1, if n is oddqn +1−2g(X1)qn/2, if n is even and p≡ 1 (mod 4)qn +1−2g(X1)qn/2, if n≡ 0 (mod 4) and p≡ 3 (mod 4)qn +1+2g(X1)qn/2, if n≡ 2 (mod 4) and p≡ 3 (mod 4).

The case p - n

Let s be the period of X1 over Fq. By Lemma 6.4.8, it follows that

s =

{2p, if p≡ 1 (mod 4)4p, if p≡ 3 (mod 4).

(6.11)

Since gcd(n,s) = 1, for each odd positive integer n such that p - n, and Nq(X1) = qp+1by Lemma 6.4.5, the sign in (6.10) is obtained as a straightforward application of Theorem 6.3.2.

Lemma 6.4.9. If p - n, then

Nqn(X1) =

qn +1, if n is even

qn +1+2g(X1)qn/2 p−1/2((−1)(n−1)/2n

p

), if n is odd

,

where(*p


Step 3: conclusion

Theorem 6.1.1 follows from Lemmas 6.4.3, 6.4.8 and 6.4.9. �


6.5 Proof of Theorem 6.1.2In this section, the proof of Theorem 6.1.2 is presented. For this, the following propo-

sition recalls two properties: the first is the well-known Quadratic Gauss Sum (see (LIDL;NIEDERREITER, 1997, Theorem 5.15)), while the second follows by the definition of µp.

Proposition 6.5.1. The following holds.

1. Let p1/2 be given as in (6.3). Then

p1/2 =p−1

∑i=0

(ip

)ζ

ip.

2. If r0,r1, . . . ,rp−1 ∈Q are such that

r0 + r1ζp + · · ·+ rp−1ζp−1p = 0,

then r0 = r1 = · · ·= rp−1.

6.5.1 Proof

By Proposition 6.3.6 and Lemma 6.4.3

LXGS(T ) =

(LX1

(T )) q−1

p−1

,

which shows that LXGS(T ) is essentially determined by LX1

(T ). Considering MX1(T ) the

reciprocal of the polynomial LX1(q−1/2T ) ∈Q(p1/2)[T ], then

MX1(T ) =

2g(X1)

∏i=1

(T −ξi),

where, for i = 1, . . . ,2g(X1), the elements ξi ∈ C satisfy

−q−n/2[

Nqn(X1)− (qn +1)]=

2g(X1)

∑i=1

ξni . (6.12)

Note that MX1(T ) is a self-reciprocal polynomial, that is, MX1

(T ) = LX1(q−1/2T ).

Further, since X1 is a supersingular curve with period s given by (6.11), the following holds

min{

n : ξni = 1 for all i = 1, . . . ,2g(X1)

}=

{2p, if p≡ 1 (mod 4)4p, if p≡ 3 (mod 4).

(6.13)

Thus, to determine the L-polynomial LX1(T ), it is necessary to describe the polynomial MX1

(T ).

From (6.13) and Lemma 6.4.8,

ξ2i ∈

{µp∪{1}, if p≡ 1 (mod 4)

µ2p∪{−1}, if p≡ 3 (mod 4),


and so2g(X1)

∑i=1

ξ2i =

{r0 + r1ζp + · · ·+ rp−1ζ

p−1p , if p≡ 1 (mod 4)

r0(−1)+ r1(−ζp)+ · · ·+ rp−1(−ζp−1p ), if p≡ 3 (mod 4),

(6.14)

with ri ∈ N being such thatp−1

∑i=0

ri = 2g(X1) = pt(p−1). Also, (6.12) and Lemma 6.4.9 imply

that2g(X1)

∑i=1

ξ2i = 0, and then the second assertion of Proposition 6.5.1 gives r0 = r1 = · · · =

rp−1 = pt−1(p−1). Thus

MX1(T ) =

2g(X1)

∏i=1

(T −ξi) =p−1

∏i=0

(T −λi)mi(T +λi)

ni, (6.15)

where mi,ni ∈ N are such that mi + ni = pt−1(p− 1), and ±λi are the square roots of the p

elements in {µp∪{1}, if p≡ 1 (mod 4)

µ2p∪{−1}, if p≡ 3 (mod 4),

for each i = 0, . . . , p−1.

Label the elements λi in a way that

λi =

{ζ i

p, if p≡ 1 (mod 4)iζ i

p, if p≡ 3 (mod 4).

Furthermore, from (6.12) and Lemmas 6.4.8, 6.4.9,2g(X1)

∑i=1

ξpi = 0 and

2g(X1)

∑i=1

ξi =−pt−1(p−1)p1/2.

Since (6.13) and Lemma 6.4.8 imply that

ξpi =

{±1, if p≡ 1 (mod 4)±i, if p≡ 3 (mod 4),

by the choice of λi and equation (6.15),

g(X1) =p−1

∑i=0

mi =p−1

∑i=0

ni (6.16)

−pt−1(p−1)p1/2 =p−1

∑i=0

(mi−ni)λi. (6.17)

In (6.17), using the first assertion of Proposition 6.5.1, one may replace p1/2 byp−1

∑i=0

(ip

)λi, if p≡ 1 (mod 4)

−p−1

∑i=0

(ip

)λi, if p≡ 3 (mod 4).

(6.18)


Hence, the second item of Proposition 6.5.1, combined with (6.16) and mi +ni = pt−1(p−1),yields n0 = m0 =

pt−1(p−1)2 and, for 16 i6 p−1,

(mi,ni) =

(0, pt−1(p−1)), if(

ip

)= 1 and p≡ 1 (mod 4)

(pt−1(p−1),0), if(

ip

)= 1 and p≡ 3 (mod 4)

(pt−1(p−1),0), if(

ip

)=−1 and p≡ 1 (mod 4)

(0, pt−1(p−1)), if(

ip

)=−1 and p≡ 3 (mod 4).

Thus, (6.15) becomes

MX1(T ) =

2g(X1)

∏i=1

(T −ξi)

=

(T 2−1

) pt−1(p−1)2

(∏

( ip )=1

(T +λi) ∏( i

p )=−1

(T −λi)

)pt−1(p−1)

, if p≡ 1 (mod 4)

(T 2 +1

) pt−1(p−1)2

(∏

( ip )=1

(T −λi) ∏( i

p )=−1

(T +λi)

)pt−1(p−1)

, if p≡ 3 (mod 4).

Let σk ∈ Gal(Q(ζp)/Q) be such that σk(ζp) = ζ kp , with 1 6 k 6 p−1. Then, the first item of

Proposition 6.5.1 implies

σk(p1/2) =

p1/2, if

(kp

)= 1

−p1/2, if(

kp

)=−1,

where p1/2 ∈Q(ζp) is defined as in (6.3). In particular,

λi/p1/2 =

ζ ip/p1/2 = σi(ζp/ p1/2), if

(ip

)= 1 and p≡ 1 (mod 4)

−ζ ip/p1/2 = σi(−ζp/ p1/2), if

(ip

)= 1 and p≡ 3 (mod 4)

ζ ip/p1/2 = σi(−ζp/ p1/2), if

(ip

)=−1 and p≡ 1 (mod 4)

−ζ ip/p1/2 = σi(ζp/ p1/2), if

(ip

)=−1 and p≡ 3 (mod 4)

6.6. Examples 139

for each i ∈ {1, . . . , p−1}. Thus, if p≡ 1 (mod 4), then

∏( i

p )=1

(p1/2T +λi) ∏( i

p )=−1

(p1/2T −λi) = p(p−1)/2∏

( ip )=1

(T +λi/p1/2) ∏( i

p )=−1

(T −λi/p1/2)

= p(p−1)/2p−1

∏i=1

(T −σi(−ζp/p1/2)

)= p(p−1)/2

∏σ∈Gal(Q(ζp)/Q)

(T −σ(−ζp/p1/2)

).

Also, for p≡ 3 (mod 4)

∏( i

p )=1

(p1/2T −λi) ∏( i

p )=−1

(p1/2T +λi) = p(p−1)/2∏

( ip )=1

(T −λi/p1/2) ∏( i

p )=−1

(T +λi/p1/2)

= p(p−1)/2p−1

∏i=1

(T −σi(−ζp/p1/2)

)= p(p−1)/2

∏σ∈Gal(Q(ζp)/Q)

(T −σ(−ζp/p1/2)

).

Therefore, the proof follows by the equalities LX1(T ) = MX1

(q1/2T ) = MX1(pt−1 p1/2T ). �

6.6 Examples

In what follows, we consider the cases p = 3,5,7,11,13,17 and 19 to illustrate Theorem6.1.2.

6.6.1 The L-polynomial of XGSfor p = 3

In this case, p≡ 3 (mod 4) and

3 ·M3(T ) = 3T 2 +3T +1,

where M3(T ) is the minimal polynomial of −ζ3/(i31/2) over Q. Therefore,

3 ·M3(3t−1T ) = qT 2 +q0T +1,

which gives that

LXGS(T ) =

((qT 2 +1) ·

(qT 2 +q0T +1

)2) q0(q−1)6

by Theorem 6.1.2.



Here p≡ 1 (mod 4) and

52 ·M5(T ) = 25T 4 +25T 3 +15T 2 +5T +1,

where M5(T ) is the minimal polynomial of −ζ5/51/2 over Q. Therefore,

52 ·M5(5t−1T ) = q2T 4 +qq0T 3 +3qT 2 +q0T +1,

which gives that

LXGS(T ) =

((qT 2−1) ·

(q2T 4 +qq0T 3 +3qT 2 +q0T +1

)2) q0(q−1)10

by Theorem 6.1.2.



73 ·M7(T ) = 343T 6 +343T 5 +147T 4 +49T 3 +21T 2 +7T +1,


73 ·M7(7t−1T ) = q3T 6 +q2q0T 5 +3q2T 4 +qq0T 3 +3qT 2 +q0T +1,

which gives that

LXGS(T ) =

((qT 2 +1) ·

(q3T 6 +q2q0T 5 +3q2T 4 +qq0T 3 +3qT 2 +q0T +1

)2) q0(q−1)14

by Theorem 6.1.2.



115 ·M11(T ) = 161051T 10 +161051T 9 +73205T 8 +14641T 7−1331T 6−1331T 5

−121T 4 +121T 3 +55T 2 +11T +1

= 115T 10 +115T 9 +5 ·114T 8 +114T 7−113T 6−113T 5−112T 4 +112T 3

+5 ·11T 2 +11T +1


115 ·M11(11t−1T ) = q5T 10 +q4q0T 9 +5q4T 8 +q3q0T 7−q3T 6−q2q0T 5−q2T 4 +qq0T 3

+5qT 2 +q0T +1

and

LXGS(T ) =

((qT 2 +1) ·

(115 ·M11(11t−1T )

)2) q0(q−1)22

by Theorem 6.1.2.

6.6. Examples 141



136 ·M13(T ) = 4826809T 12 +4826809T 11 +2599051T 10 +1113879T 9 +428415T 8

+142805T 7 +41743T 6 +10985T 5 +2535T 4 +507T 3 +91T 2 +13T +1

= 136T 12 +136T 11 +7 ·135T 10 +3 ·135T 9 +15 ·134T 8 +5 ·134T 7

+19 ·133T 6 +5 ·133T 5 +15 ·132T 4 +3 ·132T 3 +7 ·13T 2 +13T +1


136 ·M13(13t−1T ) = q6T 12 +q5q0T 11 +7q5T 10 +3q4q0T 9 +15q4T 8 +5q3q0T 7

+19q3T 6 +5q2q0T 5 +15q2T 4 +3qq0T 3 +7qT 2 +q0T +1

and

LXGS(T ) =

((qT 2−1) ·

(136 ·M13(13t−1T )

)2) q0(q−1)26

by Theorem 6.1.2.



178 ·M17(T ) = 6975757441T 16 +6975757441T 15 +3693048057T 14 +1231016019T 13

+265513259T 12 +24137569T 11−7099285T 10−4259571T 9−1252815T 8

−250563T 7−24565T 6 +4913T 5 +3179T 4 +867T 3 +153T 2 +17T +1

= 178T 16 +178T 15 +9 ·177T 14 +3 ·177T 13 +11 ·176T 12 +176T 11

−5 ·175T 10−3 ·175T 9−15 ·174T 8−3 ·174T 7−5 ·173T 6 +173T 5

+11 ·172T 4 +3 ·172T 3 +9 ·17T 2 +17T +1


178 ·M17(17t−1T ) = q8T 16 +q7q0T 15 +9q7T 14 +3q6q0T 13 +11q6T 12 +q5q0T 11−5q5T 10

−3q4q0T 9−15q4T 8−3q3q0T 7−5q3T 6 +q2q0T 5 +11q2T 4 +3qq0T 3

+9qT 2 +q0T +1

and

LXGS(T ) =

((qT 2−1) ·

(178 ·M17(17t−1T )

)2) q0(q−1)34

by Theorem 6.1.2.




199 ·M19(T ) = 322687697779T 18 +322687697779T 17 +152852067369T 16

+50950689123T 15 +15195819563T 14 +4469358695T 13 +1270238787T 12

+329321167T 11 +76759069T 10 +17332693T 9 +4039951T 8

+912247T 7 +185193T 6 +34295T 5 +6137T 4 +1083T 3 +171T 2 +19T +1

= 199T 18 +199T 17 +9 ·198T 16 +3 ·198T 15 +17 ·197T 14 +5 ·197T 13

+27 ·196T 12 +7 ·196T 11 +31 ·195T 10 +7 ·195T 9 +31 ·194T 8 +7 ·194T 7

+27 ·193T 6 +5 ·193T 5 +17 ·192T 4 +3 ·192T 3 +9 ·19T 2 +19T +1


199 ·M19(19t−1T ) = q9T 18 +q8q0T 17 +9q8T 16 +3q7q0T 15 +17q7T 14 +5q6q0T 13

+27q6T 12 +7q5q0T 11 +31q5T 10 +7q4q0T 9 +31q4T 8 +7q3q0T 7

+27q3T 6 +5q2q0T 5 +17q2T 4 +3qq0T 3 +9qT 2 +q0T +1

and

LXGS(T ) =

((qT 2 +1) ·

(199 ·M19(19t−1T )

)2) q0(q−1)38

by Theorem 6.1.2.

6.7 Proof of Theorem 6.1.3

Consider the notation in Subsections 6.3.4 and 6.3.5. Let GQ∞be the stabilizer of

Q∞ ∈XGS, G(1)

Q∞be the unique Sylow p-subgroup of GQ∞

, and let H be the cyclic complement

of G(1)Q∞

in GQ.

Let G be the set of maps on Fq(XGS) = Fq(x,y) given by

(x,y) ↦→ (αx+β ,αβq0x+α

q0+1y+ γ),

where α ∈ F*q and β ,γ ∈ Fq. One can check that G is a subgroup of G of order q2(q− 1).Moreover, the following lemma holds.

Lemma 6.7.1. G=GQ∞.

Proof. To show the inclusion G⊆GQ∞, consider the one-to-one correspondence between the

places of Fq(XGS) and the points on XGS

. In this sense, let τ be a primitive representation of


the place Q∞. Then (τ(x) : τ(y) : 1) is a primitive branch representation whose respective branchis the unique branch of GS centered at P = (0 : 1 : 0). If σ ∈G is given by

(x,y) σ↦→ (αx+β ,αβq0x+α

q0+1y+ γ),

where α ∈ F*q and β ,γ ∈ Fq, then τ ∘σ is a primitive representation for the image of Q∞ by σ .Moreover, using Theorem 6.3.15, one can verify that

(τ ∘σ(x) : τ ∘σ(y) : 1) = (ατ(x)+β : αβq0τ(x)+α

q0+1τ(y)+ γ : 1)

is also a primitive branch representation whose respective branch is a branch of GS centered at P.By the uniqueness of the branch of GS centered at P, (τ(x) : τ(y) : 1) and (τ ∘σ(x) : τ ∘σ(y) : 1)are equivalent branch representations, and therefore τ and τ ∘σ are equivalent, which impliesthat Q∞ is fixed by σ .

For the other inclusion, let σ ∈GQ∞. By Theorem 6.3.15, {1,x} and {1,x,y} are bases

for the Riemann–Roch spaces L (qQ∞) and L ((q+q0)Q∞), respectively. Therefore, σ(x) ∈L (qQ∞), σ(y) ∈L ((q+q0)Q∞),

σ(x) = αx+ γ and σ(y) = α1x+β1y+ γ1,

for some α,α1,β1,γ,γ1 ∈ Fq, with αβ1 = 0 4. Since σ is an Fq-automorphism,

0 = σ(0) = σ(yq− y− xq0(xq− x)) = σ(y)q−σ(y)−σ(x)q0(σ(x)q−σ(x))

and thus

(α1X +β1Y + γ1)q− (α1X +β1Y + γ1)− (αX + γ)q0((αX + γ)q− (αX + γ))

= λ

(Y q−Y −Xq0(Xq−X)

)(6.19)

for some λ ∈ Fq*. Therefore, comparing the coefficients in both sides of (6.19), it follows that

∙ α ∈ F*q

∙ γ ∈ Fq

∙ α1 = αγq0 ∈ Fq

∙ β1 = αq0+1 and

∙ γ1 ∈ Fq,


4 This condition is necessary for the map to be invertible.


Consider the subgroup of GQ∞consisting of the maps

(x,y) ↦→ (x+β ,β q0x+ y+ γ),

where β ,γ ∈ Fq, which is a Sylow p-subgroup of GQ∞(of order q2). By Theorem 6.3.19, this

describes the subgroup G(1)Q∞

. Also, the cyclic complement H of G(1)Q∞

in GQ∞can be given by the

maps

(x,y) ↦→ (αx,αq0+1y),

where α ∈ F*q, which has order q−1, and in this case

GQ∞=G(1)

Q∞oH.

Finally, by Theorem 6.3.20, G=GQ∞, which completes the proof. Indeed, |G(1)

Q∞|= q2 >

q0(q−1)+1 = 2g(XGS)+1, where g(XGS

) = q0(q−1)2 is the genus of XGS

, and also, sincep = 2, a comparison of genus (when t > 1) and inflection points (when m = t = 1) shows thatXGS

is not birationally equivalent to any of the curves (6.7), (6.1) and (6.8). �

6.8 Final remarksBased on (VOLOCH, 2000), this section illustrates an application of the description of

LXGS(T ) provided in Section 6.6.

It is well known (see (HIRSCHFELD; KORCHMÁROS; TORRES, 2008, Theorem9.70)) that the number of Fqn-rational points on the Jacobian variety of XGS

is given bythe number LXGS

/Fqn (1), where LXGS/Fqn (T ) is the L-polynomial of XGS

over Fqn andLXGS

/Fq(T ) = LXGS(T ).

One can check that Nq(XGS) = Nq2(XGS

) by Theorem 6.1.1, and that

LXGS(1)< LXGS

/Fq2 (1) =(((−1)

p+12 +q)(qp−1 +qp−2 + · · ·+q+1)

) q0(q−1)p

for p = 3,5,7,11,13,17 and 19. Accordingly, in these cases the group generated by the Fq2-rational points of XGS

is not the whole group of Fq2-rational points of the Jacobian variety ofXGS

, which will be denoted here as JXGS/Fq2 .

Using the construction presented in the main result of (VOLOCH, 2000) and consideringp = 3,5,7,11,13,17,19, for any subgroup G of JXGS

/Fq2 containing JXGS/Fq it is possible

to obtain an étale cover of XGSof degree i = [JXGS

/Fq2 : G] with at least i(q2 + 1) ratio-nal points over Fq2 and genus i(g− 1)+ 1, and i can be considered as being any divisor ofLXGS

/Fq2 (1)/LXGS(1).

145

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