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arXiv · 2609.17819

Only finitely many modular Jacobians are supersingular modulo a given prime

Aarya J. Kumar·Sargam Mondal·Erick Ross·Hui Xue
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Abstract

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.

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Aarya J. Kumar, Sargam Mondal, Erick Ross, Hui Xue. 2026-09-15. Only finitely many modular Jacobians are supersingular modulo a given prime. https://arxiv.org/abs/2609.17819

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