Two-cover descent on hyperelliptic curves

We describe an algorithm that determines a set of unramified covers of a given hyperelliptic curve, with the property that any rational point will lift to one of the covers. In particular, if the algorithm returns an empty set, then the hyperelliptic curve has no rational points. This provides a relatively efficiently tested criterion for solvability of hyperelliptic curves. We also discuss applications of this algorithm to curves of genus 1 and to curves with rational points.Comment: 19 pages, 1 figur

Improved rank bounds from 2-descent on hyperelliptic Jacobians

We describe a qualitative improvement to the algorithms for performing 2-descents to obtain information regarding the Mordell-Weil rank of a hyperelliptic Jacobian. The improvement has been implemented in the Magma Computational Algebra System and as a result, the rank bounds for hyperelliptic Jacobians are now sharper and have the conjectured parity

Extending techniques used to determine the set of rational points on an algebraic curve

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. 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. 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

On the Diophantine equation (nk)=(ml)+d\binom{n}{k}=\binom{m}{l}+d

By finding all integral points on certain elliptic and hyperelliptic curves we completely solve the Diophantine equation $\binom{n}{k}=\binom{m}{l}+d$ for $-3\leq d\leq 3$ and $(k,l)\in\{(2,3),\; (2,4),\;(2,5),\; (2,6),\; (2,8),\; (3,4),\; (3,6),\; (4,6), \; (4,8)\}.$ Moreover, we present some other observations of computational and theoretical nature concerning the title equation

Covering techniques and rational points on some genus 5 curves

We describe a method that allows, under some hypotheses, to compute all the rational points of some genus 5 curves defined over a number field. This method is used to solve some arithmetic problems that remained open.Comment: Contemporary Mathematics AMS, to appea