In this paper, we study supersingularity of modular Jacobians $J_0(N)$ and abelian varieties $A_f$ of $\mathrm{GL}_{2}$-type from both the vertical perspective (where the prime $p$ is fixed, and the level $N$ varies) and the horizontal perspective (where the newform $f$ is fixed, and the prime $p$ varies). Vertically, we prove that $J_{0}(N)$ is supersingular modulo a given prime $p$ for only finitely many levels $N$. Horizontally, we show that for sufficiently large $p$, the reduction of $A_f$ modulo $p$ is supersingular if and only if it isogenous to a power of a supersingular elliptic curve.
Aarya J. Kumar, Sargam Mondal, Erick Ross et al.· 0 citations
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 $S_k^{\operatorname{new}, \sigma}(N) \subseteq S_k^{\operatorname{new}}(N)$. 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 $f \in S_k^{\operatorname{new}, \sigma}(N)$ for which $p$ is extremal over $S_k^{\operatorname{new}, \sigma}(N)$. 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 $J_0^{\operatorname{new}}(N),$ $J_0^\sigma(N),$ and $J_0^{\operatorname{new}, \sigma}(N)$. 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.
Aarya J. Kumar, Sargam Mondal, Erick Ross et al.· 2 citations
Let $S_k^\sigma(N)$ denote the space of cusp forms of level $N$, weight $k$, and Atkin-Lehner sign pattern $\sigma$, and $S_k^{\mathrm{new},\sigma}$ denote its new subspace. In this paper, we study the asymptotic behavior of the coefficients of the $m$-th Hecke polynomial over $S_k^\sigma(N)$ and $S_k^{\mathrm{new},\sigma}(N)$. In particular, we show that in certain settings, all but finitely many of these coefficients take a particular sign. We also study settings in which the coefficients do not tend to any particular sign.
Timothy Nelson, Erick Ross, Maya Wassercug et al.· 0 citations
We use cookies to run the site and, with your consent, for analytics and to show ads.
See our Cookie Policy.
