Congruence properties of the coefficients of the classical modular polynomials

The classical modular polynomials Φ₍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 Φ₍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.