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.
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.