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2020•OpenGrey (Institut de l'Information Scientifique et Technique)Open access

Jacobians of hyperelliptic curves

John Paul Wunderle

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Abstract

In this thesis, we look at problems in Number Theory, specifically Diophantine Equations. We investigate Fermat Quartic curves, by presenting a set of methods to determine the existence of rational points on them. We also consider a method of resolving bielliptic curves of genus 2. We show that the method cycles for an infinite family of curves, and find an example where the method fails, however often it is repeatedly applied.

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What this paper is about

In this thesis, we look at problems in Number Theory, specifically Diophantine Equations. We investigate Fermat Quartic curves, by presenting a set of methods to determine the existence of rational points on them. We also consider a method of resolving bielliptic curves of genus 2. We show that the method cycles for an infinite family of curves, and find an example where the method fails, however often it is repeatedly applied.

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Available abstract

In this thesis, we look at problems in Number Theory, specifically Diophantine Equations. We investigate Fermat Quartic curves, by presenting a set of methods to determine the existence of rational points on them. We also consider a method of resolving bielliptic curves of genus 2. We show that the method cycles for an infinite family of curves, and find an example where the method fails, however often it is repeatedly applied.

Key concepts: Hyperelliptic curve, Hyperelliptic curve cryptography, Mathematics, Pure mathematics, Computer science, Algebra over a field, Operating system, Public-key cryptography

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