Seminar on Arithmetic Algebraic Geometry: On p-adic families of automorphic forms on GL(2) (after Hida and Coleman) (1)

2012-1-26 9:50 - 2012-1-26 11:50
Faculty of Science Building #3 Room 413
Hisaaki Kawamura
The theory of p-adic families of (elliptic) modular forms, that is, automorphic forms on GL(2) grew out of the following two important theories in arithmetic: the theory of p-adic congruences of modular forms and the theory of two-dimensional Galois representations attached to modular forms. The first example of a p-adic family is the so-called Eisenstein family consisting of non-cuspidal modular forms, considered by Serre. In this context, Hida constructed in a series of his papers published in the 1980s, p-adic families of cuspidal modular forms, which are common eigenfunctions of all Hecke operators, and in particular, possess (p-adic) unit eigenvalues of Atkin's operator U(p). Afterwards, Coleman extended Hida's construction to the case where the corresponding eigenvalue of U(p) does not necessarily admit a unit. The aim of this talk is to show you how to construct such families à la Hida-Coleman. If circumstances allow, we'd also like to introduce some recent progresses in the theory for automorphic forms on other reductive groups (ex. symplectic groups, unitary groups).