Counting congruences between modular forms in Atkin-Lehner eigenspaces

One of the arithmetically interesting operators acting on spaces of classical modular forms is the Atkin-Lehner involution, which splits these spaces into a direct sum of plus and minus eigenspaces. I will first say a bit about the classical dimension split of these eigenspaces, and then refine this story to account for congruences between modular forms (joint with Samuele Anni and Alexandru Ghitza – a collaboration begun at the MPIM!). 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 prime Hecke operators. Time permitting, I will also discuss applications of this technique towards getting partial results towards a different problem: higher congruences between p-new forms in the same weight.