A unipotent circle action on nearly overconvergent modular forms

Abstract: Recent work of Howe shows that the action of the Atkin--Serre operator on p-adic modular forms can be reinterpreted as a $\widehat{\mathbb{G}}_m$ action on the Katz Igusa tower. By p-adic Fourier theory, this gives an action of continuous functions on $\mathbb{Z}_p$ on sections of the Igusa tower (p-adic modular forms). In this talk I will explain how one can ``overconverge'' this action, i.e. show that the subspace of nearly overconvergent modular forms is stable under the action of locally analytic functions on $\mathbb{Z}_p$. This recovers (but is more general than) the construction of Andreatta--Iovita and has applications to the construction of p-adic L-functions. (Joint work with Vincent Pilloni and Joaquin Rodrigues).

algebraic geometrynumber theory

Audience: researchers in the topic

UCSB Seminar on Geometry and Arithmetic