Explicit methods for Jacobi forms over number fields

这是一个高度跨学科的建议，将数论、计算数学和表示理论联系起来。本提案旨在通过使用理论和计算方法获得对BSD猜想的新见解。BSD猜想描述了椭圆曲线的解析性质和算术性质之间的深刻关系。更确切地说，弱版本说，与椭圆曲线E相关的l函数在点1处消失的阶数（解析秩）等于由E的点（在某个给定域上）给出的阿贝尔群的秩。到目前为止，BSD猜想仅在某些类型的椭圆曲线上得到完全证明。特别地，我们知道如果解析秩小于等于1，它是成立的。然而，2014年菲尔兹奖得主M. Bhargava最近在BSD的平均版本上取得了很大的进展。我们关于BSD猜想的最广泛的知识（以及它的原始动机）来自于数值研究。目前只考虑有理椭圆曲线，也就是说，它们可以用整数系数方程来描述。通过一个著名的Eichler和Shimura的构造，我们可以将这样的椭圆曲线与某些复函数（新形式）联系起来。这种构造的逆由所谓的模性定理（原谷山-志村-韦尔猜想）给出，它在怀尔斯证明费马大定理中起了重要作用。由模性定理可知，与椭圆曲线E相关的l -函数实际上等于与E相关的新形式的l -函数。验证BSD的问题可以转化为计算模形式的l -函数，特别是它们在1处的值的问题。结果是，如果我们有一条椭圆曲线E与新形式f和L函数L（f,s）相关联，那么我们可以通过扭曲得到一整个曲线族，这样，被判别式D扭曲的曲线E就与L函数L（fxD,s）相关联。现在我们知道，存在另一个复函数F，使得F的傅立叶系数在s=1时由L（fxD,s）的值给出。如果我们计算函数F和它的傅里叶系数，我们就可以在特定的情况下验证BSD猜想。上面描述的函数F可以是半积分权值的标量新形式、向量值模形式或所谓的雅可比形式。在许多方面，后一种描述是最自然的。两个函数f和f之间的关系叫做志村对应关系。本课题研究的是椭圆曲线在数域上的BSD猜想，而不是有理数域上的BSD猜想。在这种情况下，模性定理通常是未知的，而只是推测成立。在雅可比形式下的志村对应到目前为止还没有在这个背景下被研究过。只有随着数域上的雅可比形式理论的最新发展，才有可能对这种对应应该是什么样子进行精确的推测。这个项目的主要目标是为数字域上的雅可比形式开发显式的方法和算法。特别是，我们将获得空间的维数公式，并开发允许我们计算傅里叶展开的算法，并最终获得在数域上的椭圆曲线设置中的BSD的示例。其中一个关键点是我们将Jacobi形式（在数域上）与Weil表示的向量值Hilbert模形式联系起来。以这种方式志村对应可以被实现为不同韦尔表示之间的对应。在达到上面提到的主要目标之前，我们需要更好地理解Weil表示、格和有限二次模在数域上的作用。因此，作为项目的一部分，我们也将关注这些物体。

This is a highly intra-disciplinary proposal connecting number theory, computational mathematics and representation theory. This proposal is designed to obtain new insights into the BSD conjecture through the use of theoretical and computational methods. The BSD conjecture describe a deep relationship between analytic and arithmetic properties of elliptic curves. More precisely the weak version says that the order of vanishing at the point 1, of the L-function associated with an elliptic curve E (the analytic rank) is equal to the rank of the abelian group given by the points of E (over some given field).As yet the BSD conjecture is only proven completely for certain types of elliptic curves. In particular we know that it is true if the analytic rank is less than or equal to 1. However, there has recently been a lot of progress for versions of BSD on average by the 2014 Fields medallist M. Bhargava. Our most extensive knowledge about the BSD conjecture (as well as the original motivation for it) comes from numerical investigations.Consider for the moment only rational elliptic curves, that is, they can be described by equations with integer coeffi cients. By a well-known construction of Eichler and Shimura we can associate such elliptic curves to certain complex functions (newforms). The converse of this construction is given by the so-called modularity theorem (former Taniyama-Shimura-Weil conjecture) which played an important role in Wiles' proof of Fermat's last theorem. From the modularity theorem we know that the L-function associated with the elliptic curve E is in fact equal to the L-function of the newform associated with E.The problem of verifying BSD can then be transferred to that of computing L-functions of modular forms and in particular their values at 1. It turns out that if we have an elliptic curve E with associated newform f and L-function L(f,s) then we can obtain a whole family of curves by twisting so that the curve E twisted by a discriminant D is associated to the L-function L(fxD,s).It is now known that there exists another complex function F such that the Fourier coefficients of F are given by the values of L(fxD,s) at s=1. If we compute the function F and its Fourier coefficients we might then be able to verify the BSD conjecture in specific cases.The function F described above can either be given as a scalar newform of half-integral weight, a vector-valued modular form, or a so-called Jacobi form. In many ways the latter description is the most natural one. The relationship between the two functions f and F is called the Shimura correspondence.In this project we aim at studying the BSD conjecture for elliptic curves over number fields instead of the rational numbers. In this case the modularity theorem is in general not known but only conjectured to hold. The Shimura correspondence in terms of Jacobi forms has so-far not even been studied in this setting. It is only with recent developments in the theory of Jacobi forms over number fields that it is possible to formulate precise conjectures about what the correspondence should look like.The main goal of this project is to develop explicit methods and algorithms for Jacobi forms over number fields. In particular we will obtain dimension formulas for the spaces and develop algorithms which allow us to compute Fourier expansions and in the end obtain examples for BSD in the setting of elliptic curves over number fields.One of the key points is that we associate the Jacobi forms (over number fields) with vector-valued Hilbert modular forms for the Weil representation. In this manner the Shimura correspondence can be realised as a correspondence between different Weil representations. Before reaching the main goal mentioned above we need a better understanding of Weil representations, lattices and finite quadratic modules over number fields. As part of the project we will therefore also focus on these objects.