权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Arithmetic of Automorphic Forms and Special L-Values

自守形式和特殊 L 值的算术

基本信息

批准号：
EP/N009266/1
负责人：
Athanasios Bouganis
金额：
$ 12.34万
依托单位：
Durham University
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%2FN009266%2F1
关键词：
Arithmetic Automorphic Forms Special Values

项目摘要

L-functions are known to play a central role in modern number theory. Perhaps the most well known example is the role they play in the famous Birch and Swinnerton-Dyer conjecture. Given an elliptic curve, for simplicity defined over the rational numbers, the conjecture relates various arithmetic invariants of the elliptic curve (such as the rank of the Mordell-Weil group, the size of the conjecturally finite Tate-Shafarevich group) in an astonishingly precise way to the value at s=1 of the L-function attached to the elliptic curve. This last one is defined by putting together local information, over all prime numbers, of the elliptic curve, and as defined makes sense only for the real part of s large enough (actually 3/2) and in order to make sense at s=1 one needs to establish its analytic continuation. This is now known thanks to the celebrated work of Andrew Wiles on the modularity of elliptic curves, and it is achieved by identifying the L-function of the elliptic curve with an L-function of a modular form. Actually the picture just described conjecturally extends to a much general situation. Namely to an arithmetic object, usually called a motive, one associates an L-function which it is believed to encode in its special L-values important information about the underlying motive (Bloch-Kato conjectures). However these L-functions, even though they are defined in the realm of arithmetic geometry, they can be studied with the current status of knowledge only by identifying them with the L-function of an automoprhic form. This connection between motivic and automorphic L-functions suggests that special values of automoprhic L-functions may enjoy interesting arithmetic properties. Indeed the main aim of this research project is to investigate algebraic and p-adic properties of special L-values of automorphic forms of various kinds, namely Hermitian, Siegel and Siegel-Jacobi modular forms. For Hermitian and Siegel modular forms we aim to the construction of abelian and non-abelian p-adic measures, which constitute an indispensable ingredient in the formulation of the Main Conjectures of Iwasawa Theory (commutative or not), which in turn is the only tool available to tackle the aforementioned relation implied by the Bloch-Kato conjecture between arithmetic invariants and special L-values. The situation is different with respect to the Siegel-Jacobi forms. Their L-function is not known at present to be identified to an L-function obtained from Galois representations, and actually they are not related to Shimura varieties. However they do enjoy arithmetic structure and in this research grant the aim is to address the question whether the various well-understood phenomena for automorphic forms associated to Shimura varieties (algebraicity of special L-values, Garrett's conjecture on Klingen-type Eisenstein series etc) are still valid for Siegel-Jacobi modular forms.

众所周知，L函数在现代数论中起着核心作用。也许最广为人知的例子是它们在著名的Birch和Swinnerton-Dyer猜想中所扮演的角色。给出一条椭圆曲线，为了简单地定义在有理数上，这个猜想把椭圆曲线的各种算术不变量(如Mordell-Weil群的阶，猜想有限的Tate-Shafarevich群的大小)以惊人的精确方式与附加在椭圆曲线上的L函数在S=1处的值联系起来。最后一个定义是通过将椭圆曲线上所有素数的局部信息放在一起来定义的，并且这个定义只对足够大的S的实部(实际上是3/2)有意义，并且为了在S=1处有意义，需要建立它的解析延拓。这要归功于Andrew Wiles关于椭圆曲线的模性的著名工作，它是通过用模形式的L函数来确定椭圆曲线的L函数来实现的。实际上，刚才所描述的情况只是猜想地延伸到一个非常一般的情况。也就是说，对于一个通常被称为动机的算术对象，人们将一个L函数联系起来，该函数被认为是它在其特殊的L中编码的，它重视关于潜在动机的重要信息(布洛赫-加藤猜想)。然而，尽管这些L函数是在算术几何领域中定义的，但只有将它们与自形的L函数联系起来，才能在现有的知识状况下对它们进行研究。动机函数和自同构L函数之间的这种联系表明，自同构L函数的特殊值可能具有有趣的算术性质。实际上，这个研究项目的主要目的是研究各种类型的自同构模形式，即厄米特模形式、西格尔模形式和西格尔-雅可比模形式的特殊L-值的代数性质和p-进给性质。对于厄米特模形式和西格尔模形式，我们的目标是构造阿贝尔和非阿贝尔p-进测度，它们构成了岩泽理论主要猜想(交换与否)的表述中不可或缺的组成部分，而岩泽理论又是处理布洛赫-加藤猜想所隐含的算术不变量和特殊L值之间的上述关系的唯一工具。对于Siegel-Jacobi形式，情况是不同的。它们的L函数目前还不能确定为由伽罗华表征式得到的L函数，而且实际上它们与下村变种无关。然而，它们确实具有算术结构，在这项研究中，目的是解决与Shimura簇有关的各种众所周知的自同构形式的现象(特殊L值的代数性，Garrett关于Klingen型Eisenstein级数的猜想等)是否仍然适用于Siegel-Jacobi模形式。