What can quaternions tell us about modular forms?

2:00 pm, Wednesday, 23rd October

Abstract

Modular forms and Galois representations have seen repeated successful applications to Diophantine problems.  For some applications it suffices to work with modular forms with coefficients in a finite field.  The algebraic structure of these modular forms exhibits peculiarities that are not present over the rational (or complex) numbers, mainly coming from the p-power Frobenius.  In this talk I will focus on an approach to elucidating this structure via quaternion algebras (which was discovered by Serre) and a generalisation-in-progress to the setting of Shimura curves.  This is joint work with Yiannis Fam (London School of Geometry and Number Theory).