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Mourao, Michael
Languages: English
Types: Doctoral thesis
Subjects: QA
This thesis is concerned with the problem of determining sets of rational points on algebraic curves defined over number fields. Specifically, we will explore the methods of descent, Chabauty-Coleman and the Mordell-Weil sieve. These have been around for many years, and number theorists have used them to explicitly determine the solution sets of many interesting Diophantine equations. Here we will start by giving an introduction to the basics of the existing techniques and then proceed in the second and third chapters by providing some new insights.\ud In chapter 2 we extend the method of two-cover descent on hyperelliptic curves, to the family of superelliptic curves. To do this, we need to get around some technical difficulties that arise from allowing these curves to have singular points. We show how to implement this process, and by doing this, we were able to apply descent to successfully compute the solutions to some interesting Diophantine problems, which we include in the end of the chapter.\ud Then, in chapter 3, we extend the method of "Elliptic Curve Chabauty", introduced by Bruin in [5] and independently by Flynn and Wetherell in [21], to make it applicable on higher genus curves. To fully take advantage of this technique, we combine is with a modified version of the Mordell-Weil sieve. To demonstrate the usefulness of our approach, we determine set of Q-rational points on a hyperelliptic curve of genus 6, after checking that the existing techniques could not be used to solve the same problem.
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