Congruence properties of the coefficients of the classical modular polynomials

The classical modular polynomials _ (X, Y) give plane curve models for the modular curves X₀ () /Q and have been extensively studied. In this article, we provide closed formulas for nontrivial coefficients of the classical modular polynomials _ (X, Y) in terms of the Fourier coefficients of the modular invariant function j (z) for a prime. 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.