Iwasawa Theory and Galois Representations

“该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。“该项目涉及研究数域的伽罗瓦上同调运算及其在岩泽理论中的应用。 PI已经证明了分圆p-单位上的杯积的值与以p为模的p进L-值之间的显式关系，该关系是满足与爱森斯坦级数在p以上素数处的同余的新形式。拟议的研究通过一些不同但相互交织的子项目与此相关，包括对相关模表示的塞尔默群的结构的代数研究，探索与加藤的欧拉系统和经典的主要结构之间的关系，以及某些推广的精确公式。代数数论的一个显著方面在于它发现的对象之间的联系，似乎是完全不同的性质。 这些对象可以大致分为两类：代数的和解析的。 代数对象通常是通过考虑可以通过将算术的标准运算应用于多项式方程的根而形成的数，或者通过考虑这些根的对称性来找到的。 分析对象通常是在有趣的空间上的函数，其值是复数。 该项目涉及一个意想不到的直接比较之间的代数值的一个函数对数字和解析幂级数连接到高度对称的复值函数称为模形式。 PI正在探索这一点及其在算术中的许多后果。

"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.