Arithmetic applications of Kudla-Millson theta lifts

这一提议是由朗兰兹计划所推动的，朗兰兹计划是数学家罗伯特·朗兰兹在20世纪60年代和70年代提出的一系列猜想。它们能准确预测三种看似不相关的物体之间的联系。它们来自表示理论（以模形式的形式）、数论（伽罗瓦表示）和代数几何（例如椭圆曲线）。对于这些完全不同的对象，您可以确定类似于潜在DNA的东西，即所谓的l函数。（l函数最著名的例子是“Riemann 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，对于每个模块形式，应该有一个伽罗瓦表示与相同的l -函数。建立这些联系使数学家能够更深入地了解所涉及对象的性质，并使他们能够证明定理，例如1994年怀尔斯和泰勒证明费马大定理的著名例子。自同构形式（其中包括模形式）是一类特殊的解析函数，可以研究具有不同域项的矩阵群。字段通常是一组“数字”，其中定义了加法、减法、乘法和除法的运算。一个例子是有理数领域，但除此之外还有许多其他领域。近二十年来，在有理数（和其他全实数领域）的自同构形式理论方面取得了很大进展。这导致了一些成功，比如椭圆曲线的佐藤-塔特猜想的证明。这项提议想要转移到新的领域，希望它能证明同样肥沃。新的现象出现在Bianchi模形式（虚二次域上2x2可逆矩阵的自同构形式）中，这是一个相当不同的情况，以前从代数几何中开发的工具不适用。因此，这个案例是寻找可适用于朗兰兹方案的新技术的一个重要试验场。那么我该尝试哪些技巧呢？西格尔模形式（4x4有理数上的辛矩阵）理论的最新进展使我考虑到Kudla和Millson的提升。这是一个可以用来将比安奇模形式转换为西格尔模形式的结构。我建议研究这个提升的更精细的性质，目的是应用它来回答关于比安奇模形式的问题。特别地，我想证明关于它们相关的l函数和伽罗瓦表示的结果。该提案的具体目标包括证明Bianchi模形式的l值与Kudla-Millson提振的傅立叶系数平方之间的关系（Waldspurger的一个著名公式的类似物）；证明Asai伽罗瓦表示的Bloch-Kato猜想的一个方向（解释特定l函数值的意义的结果）；以及在Calegari和Mazur推测的“p进朗兰兹泛函”的背景下研究θ升。这一建议将导致对比安奇模形式的更好理解，并将帮助从事朗兰兹方案的大型数学家社区的其他成员。

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.