The Rational Points And The Construction Of Maximal Curves Of Algebraic Curves On Finite Fields

Algebraic curves and function fields over finite fields is an important research topic in number theory and algebraic geometry. After the pioneering work of Hasse and Weil during1930s and1940s, many profound ideas have been developed and been applied to produce crucial results. In the crossroad of number theory and algebraic geometry, a new research realm called arithmetic geometry has emerged. This area now have an extensive attraction in different branches of mathematics.For a long time, algebraic curves and function fields are pure mathematical research topic. Between1977and1982, Goppa published three important papers. In these papers, algebraic geometry codes are introduced to using algebraic curves over finite fields. In Goppa’s construction, the length of algebraic geometry codes can not exceed the number of rational points on algebraic curves. Goppa’s construction unseals a new era of algebraic coding theory by number theory and algebraic geometry experts. In1985elliptic curves on finite fields provide a good candidate of one-way trap functions. Today algebraic curves over finite fields is a booming research area. It has both theoretical value in mathematics and application importance in engineering.In this thesis we will study a type of Artin-Schreier curves. By calculating some exponential sums, we obtain the number the rational points of the Artin-Schreier curves on finite fields. Moreover, by analyzing the explicit expression of the corresponding exponential sums, we obtain some maximal and minimal curves attaching Hasse-Weil bound and some maximal curves are constructed.This thesis is organized as follows:In Chapter1we introduce the background, the current development and the source of the problem and elaborate the content of this thesis.In Chapter2we will introduce basic knowledge and previous research results.In Chapter3we determine the number of rational points on a type of algebraic curves over finite fields and furthermore, construct some maximal curves.In Chapter4we make a conclusion and propose some further work.