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

这个数论研究项目研究自守形式的算术性质，自守形式是满足某些变换性质的函数。更具体地说，该项目是研究这些数学对象通过给定素数的高次幂的整除性。在过去的几年里，对这些性质的研究在数论中取得了重大进展：证明了费马大定理，证明了佐藤-泰特猜想，证明了岩泽椭圆曲线猜想。从这个项目产生的工作将提供更多的洞察一些当前最重要的问题在数论。这个项目将增强我们对p进自守形式，伽罗瓦表示及其p进L函数之间的深刻关系的认识-数论的中心焦点-并对我们对数学的理解产生重大影响。 本计画将培养研究生，研究领域为p进自守形式的演算、其伽罗瓦表示及L函数。这个项目是PI的工作的延续，该工作与各种权重和水平的自守形式之间的同余关系及其与p进L函数的联系的构建和研究有关， 塞尔默群。该项目将继续建立一些基础的一般理论的p-adic自守形式和p-adic爱森斯坦系列的眼睛一个重要的应用程序，将导致。特别是，这个理论适用于酉群和辛群的情况下，将有重要的应用，所谓的p-adic布洛赫-加藤猜想和伯奇和斯温纳顿-戴尔猜想。 研究本征簇的不可约分量的维数，近过收敛自守形式的p-adic变形，临界p-adic L-函数，自守周期的p-adic插值，附加到L-函数和Eisenstein级数的p-adic测度的构造，p-adic Euler系统和Kolyvagin系统及其与p-adic L-函数的联系是本项目将处理的一些主题。

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.