p-adic automorphic forms, p-adic L-functions and Galois representations

本课题的研究领域是p进自同构形式及其伽罗瓦表示和l函数的算法。群G的经典自同构形式是约化群G的幂点上满足良好变换性质的函数。他们的算术理论在过去几年中取得了重大进展：证明了费马大定理和佐藤-塔特猜想，仅举两个例子。上面提到的p进概念是研究这些经典对象之间的同余性的内置概念。Urban的建议是他关于爱森斯坦级数与各种权重和层次的丘形自同构形式之间的同余及其与p进l函数和某些算术定义群（称为Selmer群）的联系的工作的延续。Urban建议继续建立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，这个应用于酉群和辛群的理论将对所谓的p进布洛赫-加托猜想有重要的应用。这里是本项目将处理的几个主题的列表：(1)特征变体的不可约分量的维数(2)GL(n)的伽罗瓦表示和U（n,n）的上同调的挠类，(3)全纯和近全纯自同构形式的p进变形，(4)附在l函数和爱森斯坦级数上的p进测度的构造，(5)p进欧拉系统和p进l函数。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函数之间深层关系的认识——这是数论的一个中心焦点——并对我们对一般数学的理解产生重大影响。Urban打算写一本关于p进自同构表示的一般理论的书，以报告过去十年中已知的许多发展。本书将用一半的篇幅来阐述一般约化群的理论。在第二部分，他将用实例描述一些应用，并提出未来的研究方向。特别是，该理论的重要猜想将被详细描述。这本书的主要读者是研究生和研究人员。Urban通过组织研讨会和讲习班，与一些博士生和最近的博士后数学家合作完成他的一些项目。他还希望加强对这些年轻数学家的培训，并促进与他将定期邀请的高素质数学家的互动。这样的活动、他的图书计划以及厄本和他的学生在全国各地的其他研讨会和国际会议上的参与，将非常有助于传播该计划所取得的进展。

The domain of research of this project is the arithmetic of p-adic automorphic forms, their Galois representations and L-functions. Classical automorphic forms for a group G are functions on the adelic points of a reductive group G that satisfy nice transformation properties. Their arithmetic theory has yielded significant advances in the past few years:a proof Fermat's Last Theorem and a proof the Sato-Tate conjecture, to name two. The p-adic notions alluded above is a build-in concept to study congruences between these classical objects. Urban's proposal is a continuation on his work related to the construction and the study of congruences between Eisenstein series and cuspidal automorphic forms of various weights and levels and their links with p-adic L-functions and certain arithmetically defined groups called Selmer groups. Urban proposes to 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. Here is a list of several topics that this project will deal with:(1) Dimension of irreducible components of Eigenvarieties(2) Galois representations for GL(n) and torsion classes for the cohomology of U(n,n),(3) p-adic deformations of holomorphic and nearly holomorphic automorphic forms,(4) Construction of p-adic measures attached to L-functions and Eisenstein series,(5) p-adic Euler system and p-adic L-functions. 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. Urban intends to write a book on the general theory of p-adic automorphic representations to report on the many developments that has known the past decade. Half of the book will be devoted to the theory for general reductive group. In the second half, he will describe some applications with examples and propose directions of future research. In particular, important conjectures of the theory will be described in detail. This book will be intended mainly to graduate student and researchers. Urban works jointly with some PhD students and recent post-doctoral mathematicians on some of his projects by organizing seminar and workshops. He also wants to enhance the training of these young mathematicians and facilitate interactions with highly qualified mathematicians he will invite on a regular basis. Such an activity, his book project and the participation by Urban and his students in other seminars throughout the country and in international conferences will be very useful for the dissemination of the advances resulting from this project.