L-values, Selmer Groups, and Automorphic Forms

这一建议的焦点是数论中的中心问题之一：将L函数的特殊值与相关的代数量的阶联系起来，特别是塞尔默群。这个问题起源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。这项研究的目的是证明由自同构形产生的各种L函数之间的这种关系。自同构型理论提供了丰富的L函数(猜想全部)的来源，并通过它与与对称空间的商有关的代数簇的联系，也与代数数学家感兴趣的对象(特别是.伽罗瓦陈述)。研究人员的目标是进一步发展和利用这些联系的p-进性质来联系L-值和Selmer群。分析对象是L函数--从数论数据建立的一类特殊的分析函数(这类包括著名的由素数建立的Riemann Zeta函数)。至少一个半世纪以来，L函数一直是解决数论中最核心的问题(例如，理解素数的分布)的核心。L函数的一个重要特征是，它们在某些特殊点上的值--这些值通常被称为‘特殊值’--期望是与定义L函数的数据有关的代数量的阶。本文利用自同构型理论，证明了各类L函数的这种关系的存在性。自同构形式与分析(它们是L函数的丰富来源，推测全部)和代数都有密切的联系。研究者的目的是通过系统地理解感兴趣的对象(特定值、自同构形和Selmer群)的素数幂的可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.