The Cassels-Tate pairing for Jacobian varieties
Final Report Abstract
The goal of this project is to improve and extend methods for solving diophantine equations of the form y2 = f (x), where f is a polynomial of degree at least 5 without multiple roots and with rational coefficients, in integers or rational numbers. Equivalently, we are interested in the integral or rational points on the hyperelliptic curve defined by the equation. By a general result due to Faltings, such a curve has only finitely many rational points. It is an interesting open question whether this finite set can be determined algorithmically. There are some methods available that frequently work in practice. Most of these methods make use of the embedding of the curve into its Jacobian variety. This is an abelian variety and thus carries the helpful structure of a group. In particular, the rational points on the Jacobians form a group, the so-called Mordell-Weil group, which is known to be finitely generated. Knowledge of its free abelian rank is key for applications. 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 so 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.
