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Identifier uch.math.dip//2001magiolatidis
Title Αλγεβρικές Καμπύλες, εικασία του Riemann και κωδικοποίηση
Alternative Title Algebraic Curves, Riemann hypothesis and coding
Author Μαγιολαδίτης, Μάριος
Abstract 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 Fq and we define the group Ε(Fq) 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(Fp). The study of Ε(Fp) 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 Fq we state Riemann hypothesis for curves genus g over Fq, 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 Hasses 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.
Language Greek
Issue date 2001-11-01
Collection   School/Department--School of Sciences and Engineering--Department of Mathematics--Graduate theses
  Type of Work--Graduate theses
Permanent Link https://elocus.lib.uoc.gr//dlib/5/f/0/metadata-dlib-2001magiolatidis.tkl Bookmark and Share
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