P-adic Deformations of Eisenstein Series

埃里克·厄本奖DMS-0401131摘要。Urban将继续研究自同构形式之间的构造和同余，以攻击一些Bloch-Kato猜想和Iwasawa主要猜想，这些猜想与临界l值和Selmer群的大小有关。他将特别研究爱森斯坦级数和尖形之间的同余，比如在马祖尔-怀尔斯的作品中，或者在他关于普通椭圆曲线对称平方的主要猜想中。在这个方向上，PI将继续他与C. Skinner的联合工作，通过对U（2,2）的爱森斯坦理想的研究来证明Iwasawa关于椭圆曲线的主要猜想。他还计划指导一些博士生在一些其他单一性群体的类似主题上。PI还将通过几何和拓扑方法研究一般约化群的自同构形式的p进族的理论。这项工作将进行自己的利益，但也因为爱森斯坦级数的这种同余的研究有许多强大的应用。通过这些方法，Urban计划获得某些算术基本对象的一些新的和一般的结构，例如$p$-adic欧拉系统，它将成为约束某些Selmer群的基本工具。Urban的研究领域是自同构形式的算术理论。它们是某些具有许多对称性的全纯函数，其深层算术性质可以从它们的傅里叶系数中读出。这些物体在数论中起着至关重要的作用，因为它们的研究在过去二十年中取得了丰硕的成果，导致a .怀尔斯证明了费马大定理。本研究的最终目的是通过研究自同构形式之间的同余关系，将两个表面上不相关的对象，即l函数的特殊值（与自同构形式相关的复平面上的亚纯函数）和某些Selmer群的阶（数域的类群的推广）联系起来。Urban研究中使用的主要工具之一是p进模形式理论，该理论编码了给定素数p的任意高次模形式之间的所有同余。他将为一般约化群开发该理论的新方面，并将其应用于称为爱森斯坦级数的相关模形式，其常数项携带临界l值的信息。除其他外，本研究的结果也将对Birch和Swinnerton-Dyer的p进猜想和经典猜想产生一些重要影响，这些猜想是关于两变量三次方程解集的主要猜想

Abstract of Eric Urban's award DMS-0401131. "p-adic deformation of Eisenstein series"Urban will pursue his research on the construction and the study of congruences between automorphic forms in order to attack some of the Bloch-Kato conjectures and the Iwasawa main conjectures relating critical L-values and the size of Selmer groups. He will especially work on such congruences between Eisenstein series and cusp forms as in the works by Mazur-Wiles or in his work on the main conjecture for the symmetric square of an ordinary elliptic curve. In that direction, the PI will continue his joint work in progress with C. Skinner proving the Iwasawa main conjecture for elliptic curves via the study of the Eisenstein ideal for U(2,2). He also plans to guide some Ph-D students on a similar topic for some other unitary groups. The PI will also work on the theory of p-adic families of automorphic forms for general reductive groups by both geometric and topological approaches. This work will be undertaken for its own interest but also because the study of such congruences for Eisenstein series has numerous powerful applications. By these means, Urban plans to obtain some new and general constructions of certain arithmetical fundamental objects as for instance $p$-adic Euler systems that will be an essential tool to bound above the size of some Selmer groups.Urban's research field is the arithmetic theory of automorphic forms. Those are certain holomorphic functions having many symmetries and whose deep arithmetical properties are read from their Fourier coefficients. These objects play a fundamental role in number theory as their study has been very fruitful in the last two decades, leading A. Wiles to the proof of the Fermat's last theorem. The ultimate goal of the inverstigator's research is to relate two apparently unrelated objects which are the special values of L-functions (a meromorphic function on the complex planes associated to an automorphic form) and the order of certain Selmer groups (a generalization of the class group of a number field) via the study of congruences between automorphic forms. One of the main tools that will be used in Urban's investigation is the theory of p-adic modular forms, a theory that encodes all the congruences between modular forms modulo arbitrary high powers of a given prime number p. He will develop new aspects of this theory for general reductive groups and apply it to relevant modular forms called Eisenstein series whose constant terms carry information on critical L-values. Among other things, the results of this research will also have some important implications to the p-adic and classical conjecture of Birch and Swinnerton-Dyer that are major conjectures on the set of solutions of cubic equations in two variables