### Counting congruences between modular forms in Atkin-Lehner eigenspaces

Heilbronn Number Theory Seminar

25th October 2023, 4:00 pm – 5:00 pm

Fry Building, 2.04

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.

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