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 mod-$p$ Galois representation and Atkin-Lehner-at-$p$ sign, generalizing both recent results of Martin generalizing work of Wakatsuki and Yamauchi (no residual representation constraint) and the $\overline{\rho}$-dimension-counting formulas of Bergdall-Pollack and Jochnowitz. Working with the Atkin-Lehner involution typically requires inverting $p$, which naturally complicates investigations modulo $p$. To resolve this tension, 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, relying on our refinement of a special case of Brauer-Nesbitt, is new, purely algebraic, and may well be of independent interest. We will begin with this algebra theorem, then discuss the classical Atkin-Lehner dimension split, and only then move on to our refined dimension-counting results.

number theory

Audience: researchers in the discipline

Dublin Algebra and Number Theory Seminar

Series comments: Passcode: The 3-digit prime numerator of Riemann zeta at -11