Anna Medvedovsky

Counting congruences between modular forms in Atkin-Lehner eigenspaces

One of the arithmetically interesting operators acting on spaces of
modular forms is the Atkin-Lehner involution, which splits these
spaces into a direct sum of a plus-eigenspace and a minus-eigenspace.
I will first say a bit about the classical split in the dimensions
between these two eigenspaces, and then refine this story to account
for congruences between modular forms (joint with Samuele Anni and
Alexandru Ghitza). This is an application of a new technique for
counting mod-p modular forms recursively in the weight by establishing
deep congruences between traces of certain Hecke operators. Time
permitting, I will also discuss applications of this technique towards
getting partial results for a different problem: establishing higher
congruences between p-new forms in the same weight.