Class numbers of quadratic fields, Hasse invariants of elliptic curves, and the supersingular polynomial

Using the theory Of elliptic curves, we show that the class number h(-p) of the field Q(root-p) appears in the count of certain factors of the Legendre polynomials P-m(X) (mod p), where p is a prime > 3 and m has the form (p-e)/k, with k = 2, 3 or 4 and p equivalent to e (mod k). As part of the proof we explicitly compute the Hasse invariant of the Hessian curve y(2) + alphaxy + y = x(3) and find an elementary expression for the supersingular polynomial ss(p)(X) whose roots are the supersingular j-invariants of elliptic Curves in characteristic p. As a corollary we show that the class number h(-p) also shows Lip in the factorization (mod p) of certain Jacobi polynomials. (C) 2004 Elsevier Inc. All rights reserved.