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Arithmetic applications of Kudla-Millson theta lifts

Kudla-Millson theta 提升的算术应用

基本信息

批准号：
EP/K01174X/1
负责人：
Tobias Berger
金额：
$ 9.21万
依托单位：
University of Sheffield
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2013
资助国家：
英国
起止时间：
2013 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FK01174X%2F1
关键词：
Arithmetic applications Kudla Millson theta

项目摘要

This proposal is motivated by the Langlands programme, a series of conjectures made by the mathematician Robert Langlands in the 1960s and 70s. They predict precise links between three seemingly unrelated classes of objects. These come from representation theory (in the form of modular forms), number theory (Galois representations) and algebraic geometry (motives e.g. elliptic curves). For each of these disparate objects you can determine something akin to an underlying DNA, the so-called L-function. (The most famous example of an L-function is the "Riemann zeta function", which contains a host of arithmetic information about prime numbers). You can use this DNA to match the objects, e.g. for every modular form there should be a Galois representation with the same L-function. Establishing these links enables mathematicians to understand more deeply the properties of the objects involved and allows them to prove theorems, such as the famous example of the proof of Fermat's last theorem by Wiles and Taylor in 1994. Automorphic forms (examples of which include modular forms) are special kinds of analytic functions and can be studied for groups of matrices with entries in different fields. Fields are typically sets of "numbers" in which the operations of addition, subtraction, multiplication and division are defined. An example is the field of rational numbers but there are many other fields besides this. Much progress has been made in the theory of automorphic forms over the rational numbers (and other totally real fields) in the last two decades. This has led to successes such as the proof of the Sato-Tate conjectures for elliptic curves. This proposal wants to move to new ground in the hope that it will prove similarly fertile. New phenomena occur with Bianchi modular forms (automorphic forms for 2x2 invertible matrices over imaginary quadratic fields), a considerably different case in which previously developed tools from algebraic geometry are not applicable. This case is therefore an important testing ground for finding new techniques that could apply in the general context of the Langlands programme. So what techniques will I try? Recent progress in the theory of Siegel modular forms (4x4 symplectic matrices over the rational numbers) has led me to consider Kudla and Millson's theta lift. This is a construction that can be used to transfer Bianchi modular forms to Siegel modular forms. I propose to study the finer properties of this theta lift with the goal of applying it to answer questions about Bianchi modular forms. In particular, I want to prove results about their associated L-functions and Galois representations. Specific aims of the proposal include proving a relation between L-values of Bianchi modular forms and the squares of Fourier coefficients of the Kudla-Millson theta lifts (an analogue of a famous formula by Waldspurger); proving one direction of the Bloch-Kato conjecture for the Asai Galois representation (a result explaining the significance of the value of a particular L-function); and studying the theta lift in the context of a "p-adic Langlands functoriality" conjectured by Calegari and Mazur.This proposal will lead to a much better understanding of Bianchi modular forms and will help other members of the large community of mathematicians working on the Langlands programme.

这一提议的动机是朗兰兹纲领，这是数学家罗伯特·朗兰兹在20世纪60年代和70年代提出的一系列理论。他们预测了三个看似无关的物体类别之间的精确联系。这些来自表示论（以模形式的形式），数论（伽罗瓦表示）和代数几何（动机，例如椭圆曲线）。对于这些不同的对象中的每一个，你可以确定类似于底层DNA的东西，即所谓的L功能。(The L函数最著名的例子是"黎曼zeta函数"，它包含了大量关于素数的算术信息）。你可以使用这个DNA来匹配对象，例如，对于每个模形式，都应该有一个具有相同L函数的伽罗瓦表示。建立这些联系使数学家能够更深入地理解所涉及的对象的属性，并允许他们证明定理，例如怀尔斯和泰勒在1994年证明费马最后定理的著名例子。自守形式（其例子包括模形式）是特殊类型的解析函数，并且可以研究具有不同域中的元素的矩阵群。字段通常是一组"数字"，其中定义了加、减、乘和除的运算。一个例子是有理数领域，但除此之外还有许多其他领域。在过去的二十年里，有理数（以及其他全真实的域）上的自守形式理论取得了很大的进展。这导致了成功，如证明的Sato-Tate代数椭圆曲线。这一提议希望进入新的领域，希望它将被证明同样富有成效。新的现象出现与比安奇模形式（自守形式的2 × 2可逆矩阵的虚二次域），一个相当不同的情况下，以前开发的工具，从代数几何不适用。因此，这个案例是一个重要的试验场，寻找新的技术，可以适用于一般情况下的朗兰兹计划。那么，我应该尝试哪些技术呢？最近的进展理论的西格尔模块化形式（4x4辛矩阵的有理数）使我考虑库德拉和米尔森的θ电梯。这是一个可用于将比安奇模形式转换为西格尔模形式的构造。我建议研究更精细的性能，这θ电梯的目标是应用它来回答有关比安奇模形式的问题。特别是，我想证明有关其相关的L-函数和伽罗瓦表示的结果。该提案的具体目标包括证明比安奇模形式的L值与Kudla-Millson θ提升的傅立叶系数的平方之间的关系（Waldspurger的一个著名公式的类似物）;证明Asai Galois表示的Bloch-Kato猜想的一个方向（解释特定L函数值的重要性的结果）;并在"p进朗兰兹函数性"的背景下研究θ提升这一建议将有助于更好地理解比安奇模块化形式，并将帮助其他成员的大型社区，研究朗兰兹纲领的数学家们