Explicit methods for Jacobi forms over number fields

数字字段上雅可比形式的显式方法

基本信息

批准号：
EP/N007360/1
负责人：
Fredrik Stromberg
金额：
$ 12.42万
依托单位：
University of Nottingham
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2016
资助国家：
英国
起止时间：
2016 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FN007360%2F1
关键词：
Explicit methods Jacobi forms over

项目摘要

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.

这是连接数字理论，计算数学和代表理论的高度基于学科的建议。该建议旨在通过使用理论和计算方法来获得对BSD猜想的新见解。 BSD猜想描述了椭圆曲线的分析和算术特性之间的深厚关系。更确切地说，弱版本说，与椭圆曲线E相关的L功能（分析等级）在1点上消失的顺序等于E点（在某些给定场上）给出的Abelian群体的等级。但是，AS BSD的猜想仅在某些类型的Elliptic curves中完全证明了BSD的猜想。特别是我们知道，如果分析等级小于或等于1，这是正确的。但是，最近，2014年Fields奖牌获得者M. Bhargava的BSD版本最近有很多进展。我们对BSD猜想的最广泛的了解（以及原始动机）来自数值研究。审议目前只有理性的椭圆曲线，也就是说，它们可以通过具有整数系数的方程来描述。通过众所周知的Eichler和Shimura的结构，我们可以将这种椭圆曲线与某些复杂功能（新形式）相关联。这种结构的相反是由所谓的模块化定理（前Taniyama-shimura-Weil猜想）给出的，该定理在威尔斯的《 Fermat的最后一个定理》中发挥了重要作用。 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扭曲以使差异d扭曲的曲线与l功能l（fxd，s）有关。现在知道存在另一个复杂函数f，因此f的傅立叶系数由s = 1的l（fxd，s）值给出。如果我们计算功能F及其傅立叶系数，则可以在特定情况下验证BSD猜想。上述函数F可以作为半积分重量的标量新形式，矢量值模块化形式或所谓的Jacobi形式。在许多方面，后者的描述是最自然的描述。两个函数F和F之间的关系称为Shimura对应关系。在该项目中，我们旨在研究椭圆曲线的BSD猜想在数字字段上而不是理性数字上。在这种情况下，模块化定理通常不知道，但只有猜想要持有。在这种情况下，甚至没有研究过雅各比形式的shimura通信。只有在雅各比（Jacobi）理论上的最新发展上，数字字段上只有关于对应关系的外观的精确猜想。该项目的主要目标是开发雅各比形式的显式方法和算法，而不是数字字段。特别是，我们将获得空间的维度公式，并开发算法，使我们能够计算傅立叶扩展，并最终获得椭圆曲线的示例在椭圆曲线的设置上，而不是数字字段。一个关键点是，我们将jacobi表格（超过数字字段）与Vector-Vector-Vector-Vector-Valueld Hilbert Modull formular形式相关联。以这种方式，Shimura的对应关系可以作为不同的Weil表示之间的对应关系。在达到上面提到的主要目标之前，我们需要更好地了解Weil表示，晶格和有限的二次模块。因此，作为项目的一部分，我们还将专注于这些对象。