Explicit congruences for class equations

Abstract

Explicit congruences (mod $p$) are proved for the class equations corresponding to discriminants $D=-8p, -3p, -12p$ in the theory of complex multiplication, where $p$ is an odd prime. They are explicit in that they can be computed directly from a formula for the supersingular polynomial without first having to know the coefficients of the class equation in characteristic zero. These congruences have previously appeared in print without proof, and have been used to study factorizations of certain Legendre polynomials (mod $p$).