Université Lille 1
The eigencurve at classical weight one points
This is a joint work with Joël Bellaïche. 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 étale 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.