Counting modular forms with fixed mod-p Galois representation and Atkin-Lehner-at-p eigenvalue

Abstract: 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 generalizing work of Wakatsuki (no residual representation constraint) and the rhobar-dimension-counting formulas of Bergdall-Pollack and Jochnowitz. To resolve tension between working mod p and the need to invert p, we use the trace formula to establish up-to-semisimplifcation isomorphisms between certain mod-p Hecke modules (namely, refinements of the weight-filtration graded pieces W_k) by exhibiting ever-deeper congruences between traces of prime-power Hecke operators acting on characteristic-zero Hecke modules. This last technique is new, purely algebraic, and may be of independent interest; it relies on a combinatorial theorem whose proof benefited from a beautiful boost from Gessel.

number theory

Audience: researchers in the topic

London number theory seminar

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