We study congruences of the form F(j(z)) | U(p) = G(j(z)) mod p, where U(p)
is the p-th Hecke operator, j is the basic modular invariant
1/q+744+196884q+... for SL2(Z), and F,G are polynomials with integer
coefficients. Using the interplay between singular (a.k.a. CM) j-invariants in
characteristic zero and supersingular ones in characteristic p, we obtain such
congruences in which F is the minimal polynomial of a CM j-invariant, and give
a sufficient condition for G to be a constant polynomial in these congruences.Comment: 11 page