Iwasawa Theory and Galois Representations

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"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.