The Cassels-Tate pairing for Jacobian varieties

本项目的最终目标是改进和扩展求解形式为y^2 = f(x)的丢色图方程的方法，其中f是整数或有理数的无多根有理系数的5次或6次多项式。同样地，我们感兴趣的是由方程定义的曲线上的积分点或有理点。这种类型的曲线是2属曲线。根据Faltings的一般结果，一条属2或更大的曲线只有有限多个有理点。已知的这一事实的证明并不能得出一个在所有情况下都能确定这个有限集的算法。因此，是否存在这样的算法仍然是一个有趣的问题，而属2的曲线是研究这个问题时自然要关注的对象。在满足某些条件的情况下，有一些可行的方法在实践中起作用。这些方法大多利用了曲线可以嵌入其雅可比矩阵的事实。这是一种与曲线属相等的阿贝尔维数，因此具有群的有益结构。为了利用这种嵌入，我们需要对雅可比矩阵上的有理点群有足够的了解，这可以通过指定有限个生成器来描述。这个群被称为莫德尔-韦尔群。我们需要知道的最重要的事实是独立发电机的数量；这个数被称为莫德尔-韦尔群的秩。为了确定秩，我们在雅可比矩阵上寻找点，并检查它们的独立程度，从而得到秩的下界。我们通过计算雅可比矩阵的Selmer群来得到一个上界。这些是有限的阿贝尔群，包含一个莫德尔-韦尔群的同态象，因此知道它们的大小意味着秩的上界。不过，这个界限可能不够明确，因此，如果可能的话，能够改进它是很重要的。一种方法是找到所谓的Cassels-Tate配对的核，它是Selmer群上的双线性映射。这个核包含了莫德尔-韦尔群的图像。所以当配对是非平凡的时候我们得到了一个改进的界。为了找到核，我们必须计算Selmer群的生成子对的配对。这个项目的目标是开发一个实用的算法来计算任何给定元素对的配对值。有了这样一个算法，我们可以用它来找到一个改进的秩上界，从而在比目前可能的更多情况下确定秩，应用于上述和其他与2属曲线相关的问题。

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.