School of Mathematical Sciences

Congruences between eigensystems for GL(n)

Date(s)
Wednesday 30th October 2024 (14:00-15:00)
Contact
Event Convenor Contact: Hamid.Abban@nottingham.ac.uk
Description
Speaker's Name: Chris Williams
Speaker's Affiliation: University of Nottingham
Speaker's Research Theme(s): Pure Mathematics
Abstract:
In this talk, I will discuss congruences between modular forms. For example, consider the following question: let p be prime and f be a modular eigenform 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? The answer, amazingly, is yes. Even better, such congruences can be captured geometrically via 1-dimensional 'families' of eigenforms (via 'the eigencurve'). This theory, introduced by Hida and Coleman in the 80s/90s, has had (and continues to have) profound consequences in Iwasawa theory and the Langlands program. In this talk, my main aim will be to give a broadly accessible introduction to p-adic families, the eigencurve, and their applications to congruences of modular forms. I'll try to only assume standard knowledge of modular forms, at the level of a typical master's course. At the end I'll describe joint work with Daniel Barrera and Andy Graham, where we consider some of the problems in generalising these results to automorphic forms of GL(n) (modular forms being the case of GL(2)). Here the picture becomes more subtle -- whilst the original congruences question always has a positive answer for modular forms, in higher dimension often it is conjectured that such systematic congruences don't exist. Daniel, Andy and I study this phenomenon for congruences between symplectic forms.

Venue: Physics C04

School of Mathematical Sciences

The University of Nottingham
University Park
Nottingham, NG7 2RD

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