p-adic automorphic representations, p-adic L-functions and Bloch-Kato conjectures

本课题的目的是进一步发展p进自同构表示理论，并将该理论应用于Selmer群上的Bloch-Kato猜想等算术问题。Urban提出系统地研究任意约化群的p进自同构谱。特别是，他将p进爱森斯坦级数在这一理论中发挥核心作用，类似于朗兰兹、沙希迪等人的经典爱森斯坦级数理论。他将研究它对p进l函数研究的影响，以及p进爱森斯坦级数的泛函方程的影响。在这方面，将考虑一般群的伴生形式理论，并将其应用于某些互易律和研究p进l函数在中心临界值处的高阶消失。本课题的研究领域是自同构形式的算法。自同构形式是满足良好变换性质的全纯函数。自同构形式的算术理论是研究这些形式的傅里叶系数之间的同余。它在过去几年里取得了重大进展：证明了费马大定理，证明了佐藤-塔特猜想，举两个例子。Urban的建议是他关于特定自同构形式，爱森斯坦级数，各种权重和层次的丘形自同构形式及其与p进l函数和某些算术定义的群（称为Selmer群）之间的同余的构建和研究工作的延续。这些对象在（动机）伽罗瓦表示理论和数论中起着基本的作用。Urban提出了关于酉群和辛群的p进自同构形式和p进爱森斯坦级数的一些理论基础，目的是用它们来攻击p进bloh - kato猜想。特别是，它应该为Birch和Swinnerton-Dyer猜想提供一些新的启示，该猜想将椭圆曲线的l函数与它的一组有理解联系起来，这是数论中最著名的问题之一。

The purpose of this project is develop further the theory of p-adic automorphic representations and to give applications of this theory to arithmetic questions like the Bloch-Kato conjectures on Selmer groups. Urban proposes to study the p-adic automorphic spectrum for any reductive group in a systematic way. In particular, he will make the p-adic Eisenstein series play a central role in this theory similar to the one played by the classical theory of Eisenstein series of Langlands, Shahidi and others. He will investigate its consequences for the study of p-adic L-functions as well as the consequences of functional equations for p-adic Eisenstein series. In this aspect, the theory of companion forms for general groups will be considered with potential applications to certain reciprocity laws and to the study of high order of vanishing of p-adic L-function at central critical values.The domain of research of this project is the arithmetic of automorphic forms.Automorphic forms are holomorphic functions that satisfy nice transformation properties. The arithmetic theory of automorphic forms is the study of congruences between the Fourier coefficients of such forms. It 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. Urban`s proposal is a continuation on his work related to the construction and the study of congruences between specific automorphic forms, the 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. These objects play a fundamental role in the theory of (motivic) Galois representations and in number theory. Urban proposes to build some of the foundations of the theory of the p-adic automorphic forms and p-adic Eisenstein series for the unitary and symplectic groups with the goal of applying them to attack the p-adic Bloch-Kato conjecture. In particular, it should shed some new light on the Birch and Swinnerton-Dyer conjecture which relates the L-function of an elliptic curve to its set of rational solutions which is one of most famous problems in number theory.