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arXiv · 2406.01986

Congruence properties of the coefficients of the classical modular polynomials

Haiyang Wang
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Abstract

The classical modular polynomials $\Phi_\ell(X,Y)$ give plane curve models for the modular curves $X_0(\ell)/\mathbb{Q}$ and have been extensively studied. In this article, we provide closed formulas for $\ell$ nontrivial coefficients of the classical modular polynomials $\Phi_\ell(X,Y)$ in terms of the Fourier coefficients of the modular invariant function $j(z)$ for a prime $\ell$. Our interest in the formulas were motivated by our conjectures on congruences modulo powers of the primes $2,3$ and $5$ satisfied by the coefficients of these polynomials. We deduce congruences from these formulas supporting the conjectures.

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Haiyang Wang. 2024-06-04. Congruence properties of the coefficients of the classical modular polynomials. https://arxiv.org/abs/2406.01986

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