The Cassels-Tate pairing for Jacobian varieties

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The ultimate goal of this project is to improve and extend methods for solving diophantine equations of the form y^2 = f(x), where f is a polynomial of degree 5 or 6 without multiple roots and with rational coefficients, in integers or rational numbers. Equivalently, we are interested in the integral or rational points on the curve defined by the equation. Curves of this type are curves of genus 2. By a general result due to Faltings, a curve of genus 2 or larger has only finitely many rational points. The known proofs of this fact do not lead to an algorithm that can determine this finite set in all cases. So it remains an interesting question whether such an algorithm exists, and curves of genus 2 are the natural objects to focus on when studying this question.There are some methods available that work in practice when some conditions are satisfied. Most of these methods make use of the fact that the curve can be embedded into its Jacobian variety. This is an abelian variety of dimension equal to the genus of the curve and thus carries the helpful structure of a group. To make use of this embedding, we need to know enough about the group of rational points on the Jacobian variety, which can be described by specifying finitely many generators. This group is known as the Mordell-Weil group. The most important fact we need to know is the number of independent generators; this number is called the rank of the Mordell-Weil group.To determine the rank, we search for points on the Jacobian variety and check to what extent they are independent, thus obtaining a lower bound on the rank. We obtain an upper bound by computing so-called Selmer groups of the Jacobian. These are finite abelian groups containing a homomorphic image of the Mordell-Weil group, and so knowing their size implies an upper bound for the rank.This bound may fail to be sharp, though, and so it is important to be able to improve it if possible. One way of doing this is to find the kernel of the so-called Cassels-Tate pairing, which is a bilinear map on the Selmer group. This kernel contains the image of the Mordell-Weil group. So we get an improved bound when the pairing is nontrivial. To find the kernel, we have to evaluate the pairing on pairs of generators of the Selmer group. The goal of this project is to develop a practical algorithm that computes the value of the pairing on any pair of given elements. Having such an algorithm at our disposal, we can use it to find an improved upper bound for the rank and thus determine the rank in many more cases than currently possible, with applications as described above and to other questions related to curves of genus 2.