Department of Mathematics,

University of Crete

 

 

 

 

 

Algebraic Curves,

Riemann hypothesis and coding

 

 

Marios Magioladitis

 

 

 

Diploma Thesis

 

 

 

 

 

 

 

 

 

 

 

 

Heraklion, 2001

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This essay was submitted on the Department of Mathematics of the University of Crete on November 2001. The supervisor was professor Jannis A. Antoniadis.

 

The evaluation commitee consisted of Jannis Antoniadis, Alexis Kouvidakis and Aristides Kontogeorgis.

 

 

Contents

 

 

 

Introduction	3
CHAPTER I	5
Elements of Algebraic Curve Theory
1.      Projective plane	5
2.      Plane algebraic curves	7
3.      Intersection points of algebraic curves and their multiplicity	9
4.      Singular points (inflections or flexes)	15
5.      Elliptic curves	19
CHAPTER II	23
Cubic Curves Over Finite Fields
1.      Rational points over cubic curves	23
2.      Points of finite order	27
3.      The Riemann Hypothesis	37
4.      Manin’s Proof of the Hasse Inequality	41
5.      Proof of the Basic Identity	51
CHAPTER III	61
Algebraic Curves and Coding Theory
1.      Elements of coding theory	61
2.      Codes defined using algebraic curves	71
3.      Examples of algebraic geometry codes	75
Bibliography	79
Index	81

 

Introduction

 

 

The purpose of this essay is to show the usefulness of studying algebraic curves over finite fields, as far as Number Theory problems and Coding Theory are concerned.

 

In the first chapter we discuss basic properties of the theory, for example the intersection points of algebraic curves and their multiplicity. Furthermore, we define elliptic curves over the field Q of rational numbers and we state important theorems, which concern the group of their rational points.

 

In the second chapter we discuss elliptic curves over finite fields in the form F_q and we define the group Ε(F_q) of their rational points. Furthermore, it is shown that if we have an elliptic curve defined on Q, with integer coefficients and we reduce it modulo p for proper primes p then the group Φ of the rational points with finite order of E is isomorphic to a subgroup of Ε(F_p). The study of Ε(F_p) give us useful information  for Φ as well. In order to give an upper bound for the number of rational points of an algebraic curve defined over F_q we state Riemann hypothesis for curves genus g over F_q, while we introduce thoroughly Manin’s Proof of the Hasse Inequality which is a special case of Riemann Hypothesis for curves. We also explain it's relation with the famous Riemann Hypothesis and the ζ-eta function. The chapter ends with a letter from Mr. Roquette to Mr. Lemmermeyer, which proves that Manin’s proof and Hasse’s proof are, in fact, similar.

 

In the third chapter we state basic notions of coding theory and give some bounds for the efficiency of the codes that we can construct. Moreover we give some additional elements of algebraic curves theory and coding theory, for example we define the divisor of a curve and the algebraic Reed-Solomon codes. Finally, it is shown if we consider algebraic curves with a lot of rational points we can construct efficient codes. The essay ends with the detailed presentation of two algebraic geometry codes.

 

I would like to thank professor Jannis A. Antoniadis for his valuable help. He revealed me a wonderful branch of Mathematics and due to his persistence this essay has been completed.

 

Best regards to professor Mr. Peter Roquette (Heidelberg) who permitted me to include his letter in my essay, as well as to professor Mr. Ruud Pellikaan (Eindhoven) for his help in some coding theory problems.