Congruence properties of the coefficients of the classical modular polynomials

The classical modular polynomials ${\mathrm{\Phi }}_{\mathrm{\ell }}(X,Y)$ give plane curve models for the modular curves $X_0(\mathrm{\ell })\mathrm{/}\mathbb{Q}$ and have been extensively studied. In this article, we provide closed formulas for $\mathrm{\ell }$ nontrivial coefficients of the classical modular polynomials ${\mathrm{\Phi }}_{\mathrm{\ell }}(X,Y)$ in terms of the Fourier coefficients of the modular invariant function $j(z)$ for a prime $\mathrm{\ell }$. 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.