p-Adic Families of Modular Forms
An elliptic modular form (of some weight and level) is a complex
valued holomorphic function on the upper half plane which satisfies
a certain transformation property with respect to the action of a
subgroup of SL(2, Z) and which is also holomorphic at the boundary
of the upper half plane. To some of these modular forms (more
precisely to the eigenforms for the Hecke operators) one can attach
a complex L-function and a family of representations of the absolute
Galois group of the rationals.
If E is an elliptic curve over the rationals, we say that E is
modular if the L-function of E (or, equivalently, the family of
Galois representations attached to E) equals the L-function
(respectively, the Galois representations) of a weight 2 modular
eigenform of a precisely defined level.
In 1996 Andrew Wiles and his collaborators proved that every
elliptic curve over the rationals is modular and this implied the
famous Fermat's Last Theorem.
In this talk we will be particularly interested in congruences
modulo integers between various modular forms. R. Coleman noticed in
the congruences modulo powers of a fixed prime p between elliptic
modular forms are explained by the existence of certain p-adic
families of such objects. Recently, together with F. Andreatta, V.
Pilloni and G. Stevens we have been able to give geometric
Coleman's constructions and to generalize them to Hilbert and Siegel