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

Abstract

Using the theory of elliptic curves, we show that the class number h(− p) of the field Q ( −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≡e ( mod k) . As part of the proof we explicitly compute the Hasse invariant of the Hessian curve y 2+ αxy+ 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 up in the factorization ( mod p) of certain Jacobi polynomials.