Congruences between eigensystems for GL(n)

Abstract: In this talk, I will discuss congruences between modular forms, and -- more generally -- between systems of eigenvalues on GL(n), modular forms being the special case $n=2$. In a broad sense, this seeks to answer the following type of question: Let $p$ be prime and $f$ be a modular form of level $\Gamma_0(M)$ where $p$ divides $M$. For a given integer $m$, does there exist another eigenform $g$ congruent to $f$ modulo $p^m$?

For modular forms, the answer is yes. Even better, such congruences can be captured geometrically via 1-dimensional families' of eigenforms (via the eigencurve'). This geometric object has had profound consequences in Iwasawa theory and the Langlands program. In this talk, I will attempt to give a gentle introduction to $p$-adic families, and describe joint work with Daniel Barrera and Andy Graham, where we consider some of the problems in generalising them to higher dimension. Here the picture becomes more subtle -- whilst all families for GL(2) are $1$-dimensional, for GL(4), there can be classical families of dimension $0, 1$, or $2$ attached to the same automorphic representation.

Mathematics

Audience: researchers in the topic

ANTLR seminar

Series comments: This is the Algebra, Number Theory, Logic and Representation theory seminar.