Congruence properties of the coefficients of the classical modular polynomials

TL;DR

This paper derives explicit formulas for coefficients of classical modular polynomials and investigates their congruence properties modulo powers of small primes, supporting conjectures about these congruences.

Contribution

Provides closed-form formulas for nontrivial coefficients of classical modular polynomials in terms of Fourier coefficients of the j-invariant, and explores their prime power congruences.

Findings

Formulas for coefficients in terms of Fourier coefficients of j(z)

Evidence supporting conjectures on prime power congruences

Insights into the arithmetic structure of modular polynomial coefficients

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.