Overconvergent Differential Operators for Hilbert Modular Forms

Abstract: In the 1970's, Katz constructed p-adic L-functions for CM fields by relating the values of the Dedekind zeta function to the values of certain nearly holomorphic Eisenstein series. Crucial in his construction was the action of the Maass--Shimura differential operators. Katz's p-adic interpolation of these differential operators is only defined over the ordinary locus, which leads to a restriction on what p are allowed. Recently, this restriction has been lifted in the case of quadratic imaginary fields by Andreatta and Iovita using an "overconvergent" analog of the Maass--Shimura operator for elliptic modular forms. We will give an overview of the theory of overconvergent Hilbert modular forms before constructing an "overconvergent" analog of the Maass--Shimura operator for this setting.

algebraic geometrynumber theory

Audience: researchers in the topic

UCSB Seminar on Geometry and Arithmetic