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arXiv · 2509.06423

Divisibility of the coefficients of modular polynomials

Florian Breuer
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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.

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Florian Breuer. 2025-09-08. Divisibility of the coefficients of modular polynomials. https://arxiv.org/abs/2509.06423

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