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October 3, 2025Open Access

Divisibility of the coefficients of modular polynomials

Key Points

The coefficients of modular polynomials are highly divisible by small primes, especially at certain parameters.
For algebraic numbers like J=0 and J=1728, unique divisibility properties are revealed.
Cyclic isogenies of degree N create vanishing pairs in modular polynomials, impacting coefficient divisibility.
Understanding these divisibility patterns can inform studies in number theory and elliptic curves.

Abstract

Let N>1 and let ΦN (X, Y), 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 ΦN (X+J, Y+J) for certain algebraic numbers J, in particular J=0 and J=1728. It turns out that these coefficients are highly divisible by small primes at which J is supersingular.

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Florian Breuer (Mon,) studied this question.

synapsesocial.com/papers/68e02f3cf0e39f13e7fa27a4 https://doi.org/https://doi.org/10.48550/arxiv.2509.06423

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Divisibility of the coefficients of modular polynomials

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Abstract

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