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

本数论研究项目研究了自同构型的算术性质，自同构型是满足一定变换性质的函数。更具体地说，这个项目是研究这些数学对象的给定素数的高次幂的整除性质。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：费马最后定理的证明，佐藤-塔特猜想的证明，以及关于椭圆曲线的岩泽主要猜想的证明。这个项目所产生的工作将使我们更深入地了解当前数论中的一些最重要的问题。这个项目将加强我们对p-进自同构形、伽罗瓦表示及其p-进L函数之间的深层关系的了解--这是数论的一个中心焦点--并对我们对一般数学的理解有重要的影响。该项目将涉及研究生的培养。该项目的研究领域是p-进自同构形的算术，它们的伽罗瓦表示，以及L函数。本项目是PI关于不同权重和层次的自同构型之间的同余以及它们与p-进L函数和Selmer群的联系的工作的继续。该项目将继续建立p-进自同构形和p-进Eisenstein级数的一般理论的一些基础，并着眼于将产生的重要应用。特别是，这一理论应用于酉群和辛群的情形，将对所谓的p-进Bloch-Kato猜想和Birch和Swinnerton-Dyer猜想有重要的应用。研究本征簇的不可约分量的维度、几乎过收敛自同构型的p-adic变形、临界p-adgeo-L-函数、自同构周期的p-adic插值、L函数和Eisenstein级数的p-adic测度的构造、p-adic Euler系和Kolyvan in系统及其与p-ad场L-函数的联系是本课题要研究的内容之一。

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.