Work in progress joint with Samuele Anni and Alexandru Ghitza. For N
prime to p, we count the number of classical modular forms of level Np
and weight k with fixed residual Galois representation and
Atkin-Lehner-at-p sign, generalizing both recent results of Martin (no
residual representation constraint) and rhobar-dimension-counting
formulas of Jochnowitz and Bergdall-Pollack. One challenge is the
tension between working modulo p and the need to invert p when working
with the Atkin-Lehner involution. To address this, we use the trace
formula to establish up-to-semisimplifcation isomorphisms between
certain mod-p Hecke modules (namely, refinements of weight-graded
pieces of spaces of mod-p forms) by exhibiting ever-deeper congruences
between traces of prime-power Hecke operators acting on
characteristic-zero Hecke modules. This last technique is new and
combinatorial in nature; it relies on a theorem discovered by the
authors and beautifully proved by Gessel, and may be of independent
interest.