Weight one forms in p-adic families

A p-adic family of ordinary eigenforms contains infinitely many classical members of any fixed weight at least 2. Moreover, every classical p-ordinary eigenform of weight at least 2 is known to live in a unique family. However, non-CM families contain only finitely many weight 1 members, and moreover, one expect the uniqueness result to fail in weight 1. In this talk we shall a) give sharp estimates for the exact number of weight 1 members in families and b) give recipies and examples to show how exactly uniqueness fails. This is joint work with Mladen Dimitrov.