**Haruzo Hida**

UCLA, USA

July 16, 2015

**Non CM p-adic analytic families of modular forms**:
A p-adic analytic family is a family of modular forms f_P
(eigenforms of all Hecke operators) parameterized by geometric points P of an integral scheme Spec (I) finite flat over Spec (Z_p[[T]]) whose geometric points is an open unit disk of the p-adic field. The Hecke eigenvalues of T(n) of f_P is given by the value a_n(P) for a_n in the structure sheaf I (so analytic). Such a family has corresponding Galois representation r_I into GL_2(I). The family is said to have complex multiplication if r_I has essentially an abelian image. In this talk, we emphasize by examples importance of characterizing CM and non CM families.