Number Theory Seminar

This is a joint work with Joel Bellaiche. We determine the
geometry of the p-adic eigencurve at points corresponding to classical
modular forms of weight one, under a mild assumption of regularity at p,
and give several number theoretic applications. Namely we prove that the
eigencurve is always smooth at those points, and that it is etale over the
weight space if, and only if, the form does not have real multiplication
by a real quadratic field in which p splits. Our approach uses deformation
theory of Galois representations and the Baker–Brumer theorem in
transcendence theory.