Effective Hecke eigenvalue equidistribution over the Atkin--Lehner subspaces

Abstract

For a fixed prime $p$, let ${\mu }_p$ denote the $p$-adic Plancherel measure. Then the first main goal of this paper is to prove effective (and moreover explicit) ${\mu }_p$-equidistribution of the $p$-th Hecke eigenvalues over the Atkin--Lehner subspaces $S_k^{\sigma }(N)\subseteq S_k(N)$ and $S_k^{\mathrm{new},\sigma }(N)\subseteq S_k^{\mathrm{new}}(N)$. We then highlight five applications of this explicit equidistribution result. For the first application, we generalize Kim's vertical analog of the Atkin--Serre conjecture to the Atkin--Lehner setting. For the second application, we obtain explicit bounds on the number of newforms $f\in S_k^{\mathrm{new},\sigma }(N)$ for which $p$ is extremal over $S_k^{\mathrm{new},\sigma }(N)$. For the third application, we prove explicit asymptotics for the number of ${\mathbb{F}}_{p^r}$-points on the modular Jacobian $J_0(N),$ as well as on its factors $J_0^{\mathrm{new}}(N),$ $J_0^{\sigma }(N),$ and $J_0^{\mathrm{new},\sigma }(N)$. We also make explicit an asymptotic result of Serre concerning point counts of the modular curves $X_0(N)$. For the fourth application, we generalize lower bounds due to Murty and Sinha on the sizes of large $\mathbb{Q}$-simple factors of $J_0(N)$ to analogous bounds for $J_0^{\sigma }(N)$. Finally, for the fifth application (the details of which are given in a separate paper), we use our explicit equidistribution result to prove that only finitely many modular Jacobians are supersingular modulo any fixed prime.

52 pages