Effective Hecke eigenvalue equidistribution over the Atkin--Lehner subspaces

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.