Iwasawa Theory and Galois Representations

“这项奖励是根据2009年美国复苏和再投资法案（公法111-5）资助的。”本课题研究数域伽罗瓦上同的运算及其在Iwasawa理论中的应用。PI已经推测出环切p单位上的杯积的值与p进l值之间的明确关系，取模p，满足在p以上的素数处与爱森斯坦级数同余的新形式。提议的研究通过许多不同但相互交织的子项目与此相关，包括对相关模表示的Selmer群结构的代数研究。探索与加藤欧拉系统和经典主要猜想的关系，以及某些推广的精确公式。代数数论的一个引人注目的方面在于它发现了似乎具有完全不同性质的物体之间的联系。这些对象大致可以分为两类：代数对象和解析对象。代数对象通常是通过考虑可以通过对多项式方程的根应用标准算术运算或考虑这些根的对称性而形成的数字来找到的。解析对象通常是有趣空间上的函数，其值为复数。该项目涉及到一种意想不到的直接比较，即在数对上的函数的代数值与高度对称的复值函数（称为模形式）的解析幂级数之间进行比较。PI正在探索这一点及其在算术上的许多影响。

"This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5)." The project involves a study of operations in the Galois cohomology of number fields and their application in Iwasawa theory. The PI has conjectured an explicit relationship between the values of a cup product on cyclotomic p-units and p-adic L-values, taken modulo p, of newforms that satisfy congruences with Eisenstein series at a prime above p. The proposed research relates to this through a number of distinct but intertwined sub-projects, including an algebraic study of the structure of the Selmer groups of the associated modular representations, the exploration of relationships with Kato's Euler system and classical main conjectures, and the precise formulation of certain generalizations.A remarkable aspect of algebraic number theory lies in the connections it finds between objects that appear to be of entirely different natures. These objects can roughly be described as falling into two classes: those that are algebraic, and those that are analytic. The algebraic objects are typically found by considering numbers that can be formed by applying the standard operations of arithmetic to the roots of polynomial equations, or by considering the symmetries of those roots. The analytic objects are often functions on interesting spaces with values that are complex numbers. The project concerns an unexpected direct comparison between the algebraic values of a function on pairs of numbers and the analytic power series attached to highly symmetrical complex-valued functions known as modular forms. The PI is exploring this and its many consequences in arithmetic.