The Cassels-Tate pairing for Jacobian varieties

这个项目的最终目标是改进和推广求解y^2=f(X)形式的丢番图方程的方法，其中f是5次或6次多项式，没有重根，具有有理系数，为整数或有理数。同样，我们对方程所定义的曲线上的积分点或有理点感兴趣。这类曲线是亏格2的曲线。根据Faltings的一般结果，亏格为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导致在所有情况下都能确定该有限集的算法。因此，是否存在这样的算法仍然是一个有趣的问题，而亏格2的曲线是研究这个问题的自然对象。在满足某些条件的情况下，有一些可用的方法在实践中起作用。这些方法中的大多数都利用了曲线可以嵌入到其雅可比变换中这一事实。这是一种维度与曲线亏格相等的阿贝尔变种，因此承载着有用的群体结构。为了利用这种嵌入，我们需要充分了解雅可比簇上的有理点群，这可以通过指定有限多个生成元来描述。这个群被称为莫代尔-韦尔群。我们需要知道的最重要的事实是独立生成元的数目，这个数目称为Mordell-Weil群的秩.为了确定秩值，我们在Jacobian簇上寻找点，并检查它们在多大程度上是独立的，从而得到秩下界.通过计算雅可比的所谓Selmer群，我们得到了一个上界。这些群是有限阿贝尔群，包含Mordell-Weil群的同态像，因此知道它们的大小意味着它的一个上界.这个上界可能不是尖锐的，所以如果可能的话，能够改进它是很重要的.一种方法是找到所谓的Cassel-Tate配对的核，它是Selmer群上的双线性映射。该内核包含Mordell-Weil组的图像。因此，当配对是非平凡的时，我们得到了一个改进的界。为了找到核，我们必须计算Selmer群的生成元对上的配对。这个项目的目标是开发一个实用的算法，计算任何一对给定元素的配对的值。有了这样的算法，我们可以使用它来找到改进的秩上界，从而在比目前可能的情况下更多地确定秩且如上所述地应用于与亏格2的曲线有关的其他问题。

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.