p-adic automorphic forms, p-adic L-functions, and Selmer groups

本数论课题研究自同构形式的算术性质，自同构形式是满足一定变换性质的函数。更具体地说，该项目是研究这些数学对象被给定素数的高幂的可整除性。在过去的几年中，对这些性质的研究在数论方面取得了重大进展：证明了费马大定理，证明了佐藤-塔特猜想，证明了椭圆曲线的岩泽主猜想。从这个项目中产生的工作将使我们更深入地了解目前数论中一些最重要的问题。该项目将增强我们对p进自同构形式、伽罗瓦表示及其p进L函数（数论的中心焦点）之间深层关系的认识，并对我们对一般数学的理解产生重大影响。这个项目将包括研究生的培训。本课题的研究领域是p进自同构形式的算术，它们的伽罗瓦表示和l函数。该项目是PI关于各种权重和层次的自同构形式及其与p进l函数和Selmer群的联系之间的同余的构建和研究工作的延续。该项目将继续建立p进自同构形式和p进爱森斯坦级数的一般理论的一些基础，并着眼于将产生的重要应用。特别地，这一理论应用于酉群和辛群的情况将对所谓的p进布洛赫-卡托猜想和伯奇和斯温纳顿-戴尔猜想有重要的应用。研究特征变不可约分量的维数、近过收敛自同构形式的p进变形、临界p进l函数、自同构周期的p进插值、l函数和爱森斯坦级数的p进测度的构造、p进欧拉系统和Kolyvagin系统及其与p进l函数的联系是本课题将涉及的一些主题。

This research project in number theory studies arithmetic properties of automorphic forms, which are functions that satisfy certain transformation properties. More specifically, the project is to study divisibility properties of these mathematical objects by high powers of a given prime number. The study of these properties has yielded significant advances in number theory in the past few years: a proof Fermat's Last Theorem, a proof the Sato-Tate conjecture, and a proof of the Iwasawa main conjecture for elliptic curves. The work resulting from this project will give more insight into some of the current most important problems in number theory. This project will enhance our knowledge of the deep relationships between p-adic automorphic forms, Galois representations, and their p-adic L functions -- a central focus of number theory -- as well as have significant consequences for our understanding of mathematics in general. The project will involve the training of graduate students.The domain of research of this project is the arithmetic of p-adic automorphic forms, their Galois representations, and L-functions. This project is a continuation of the PI's work related to the construction and the study of congruences between automorphic forms of various weights and levels and their links with p-adic L-functions and Selmer groups. The project will continue to build some of the foundations of the general theory of the p-adic automorphic forms and p-adic Eisenstein series with an eye one the important applications that will result. In particular, this theory applied to the case of unitary and symplectic groups will have important applications to the so-called p-adic Bloch-Kato conjecture and Birch and Swinnerton-Dyer conjecture. Studying the dimension of irreducible components of eigenvarieties, p-adic deformations of nearly over-convergent automorphic forms, critical p-adic L-functions, p-adic interpolation of automorphic periods, construction of p-adic measures attached to L-functions and Eisenstein series, p-adic Euler systems, and Kolyvagin systems and their link with p-adic L-functions are some of the topics that will be dealt with in this project.