L-values, Selmer Groups, and Automorphic Forms

这一建议的焦点是数论中的一个中心问题：将l函数的特殊值与相关代数量的阶，特别是Selmer群的阶相关联。这个问题的根源在于代数数论中著名的类数公式，其中包括Birch和Swinnerton-Dyer关于椭圆曲线的l函数的猜想。本文的研究旨在证明由自同构形式产生的各种l函数的这种关系。自同构形式理论提供了l函数的丰富来源（推测上全部），并且-通过它与对称空间商的代数变异的联系-它也与代数数论家感兴趣的对象（特别是伽罗瓦表示）密切相关。研究者的目标是进一步发展和利用这些连接的p进性质来联系l值和Selmer群。本提案中描述的研究旨在建立某些解析对象和代数对象之间的关系。解析对象是l函数——一类由数论数据构建的特殊解析函数（这类函数包括著名的黎曼ζ函数，它是由素数构建的）。至少一个半世纪以来，l函数一直是解决数论中最核心问题（例如，理解素数的分布）的核心。l函数的一个重要特征是它们在某些特殊点的值——这些值通常被称为“特殊值”——被期望是与定义l函数的数据相关的代数量的阶数。研究者的目的是证明这种关系的存在对各种类型的l -函数，特别是借鉴自同构形式的理论。自同构形式与分析（它们是l函数的丰富来源，推测上全部）和代数密切相关。研究者的目标是通过系统地理解感兴趣的对象（特殊值、自同构形式和塞尔默群）的幂素数的可整除性来证明这种关系。

The focus of this proposal is one of the central problems in numbertheory: relate special values of L-functions to the orders of associated algebraic quantities, especially Selmer groups. This is a problem whose origins lie in the celebrated class number formula from algebraic number theory and which includes the conjecture of Birch and Swinnerton-Dyer about the L-function of an elliptic curve. The research described in this proposal aims to prove such relations for various L-functions arising from automorphic forms. The theory of automorphic forms provides a rich source of L-functions (conjecturally all) and - through its connections with algebraic varieties attached to quotients of symmetric spaces - it is also closely connected to the objects of interest to algebraic number theorists (esp. Galois representations). The investigator aims to further develop and exploit the p-adic properties of these connections to relate L-values and Selmer groups.The research described in this proposal aims to establish relations between certain analytic and algebraic objects. The analytic objects are L-functions- a special class of analytic functions built from number-theoretic data (this class includes the celebrated Riemann zeta function which is built from the prime numbers). For at least a century and a half L-functions have been central to efforts to tackle the most central problems in number theory (e.g., understanding the distribution of prime numbers). An important feature of L-functions is that their values at certain special points - these values are often called `special values' - are expected to be the orders of algebraic quantities associated to the data defining the L-function. The investigator aims to prove the existence of such relations for various classes of L-functions, drawing especially on the theory of automorphic forms. Automorphic forms are closely connected to both analysis (they are a rich source of L-functions, conjecturally all) and algebra. The investigator aims to prove such relations by systematically understanding the divisibility properties of the objects of interest (special values, automorphic forms, and Selmer groups) by powers of primes numbers.