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Divisibility of the coefficients of modular polynomials
[Submitted on 8 Sep 2025 (v1), last revised 25 Jun 2026 (this ve · 2026-06-26 · via math updates on arXiv.org

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Abstract:Let $N>1$ and let $\Phi_N(X,Y)\in\mathbb{Z}[X,Y]$ be the modular polynomial which vanishes precisely at pairs of $j$-invariants of elliptic curves linked by a cyclic isogeny of degree $N$. In this note we study the divisibility of the coefficients of $\Phi_N(X+J, Y+J)$ for certain algebraic numbers $J$, in particular $J=0$ and other singular moduli. It turns out that these coefficients are highly divisible by small primes at which $J$ has supersingular reduction.

Submission history

From: Florian Breuer [view email]
[v1] Mon, 8 Sep 2025 08:18:11 UTC (13 KB)
[v2] Thu, 16 Oct 2025 12:50:24 UTC (23 KB)
[v3] Thu, 25 Jun 2026 05:01:05 UTC (17 KB)