Arithmetic of singular moduli and class polynomials

Abstract

We investigate divisibility properties of the traces and Hecke traces of singular moduli. In particular we prove that, if p is prime, these traces satisfy many congruences modulo powers of p which are described in terms of the factorization of p in imaginary quadratic fields. We also study generalizations of Lehner’s classical congruences j(z)|Up ≡ 744 (mod p )( wherep 11 and j(z) is the usual modular invariant), and we investigate connections between class polynomials and supersingular polynomials in characteristic p.