Endoscopic p-adic modular forms for SL(2)

Abstract: An important question in the theory of $p$-adic modular forms is to recognize classical modular forms in the vast sea of $p$-adic modular forms. For example, for GL(2) over $\Q$, a $p$-adic overconvergent modular eigenform whose Hecke eigenvalues agree with those of a classical eigenform is in fact a classical eigenform. Judith Ludwig discovered, by a non-constructive method, that this need not be the case for SL(2). The goal of my talk will be to explain how to understand and quantify this phenomenon using ideas from the geometry of "moduli spaces of Galois representation". Along the way we also obtain results on the local geometry of SL(2)-eigenvarieties at endoscopic classical points. This is joint work in progress with Judith Ludwig.

algebraic geometrynumber theory

Audience: researchers in the topic

Séminaire de géométrie arithmétique et motivique (Paris Nord)