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

2012126 9:50 
2012126 11:50
 Place
 Faculty of Science Building #3 Room 413

Speaker/Organizer
 Hisaaki Kawamura

 The theory of padic 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 padic congruences of modular forms and the theory of twodimensional Galois representations attached to modular forms. The first example of a padic family is the socalled Eisenstein family consisting of noncuspidal modular forms, considered by Serre. In this context, Hida constructed in a series of his papers published in the 1980s, padic families of cuspidal modular forms, which are common eigenfunctions of all Hecke operators, and in particular, possess (padic) 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 HidaColeman. 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).