Arithmetic of singular moduli and class polynomials
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 $j(z)| U_p\equiv 744 \pmod p$ (where $p\leq 11$ and j(z) is the usual modular invariant), and we investigate connections between class polynomials and supersingular polynomials in characteristic p.(Received April 29 2003)
(Accepted May 14 2004)
(Published Online February 10 2005)
Key Words: singular moduli; class polynomials; modular forms.
11F33; 11F37 (primary).