Legendre polynomials and complex multiplication, I

The factorization of the Legendre polynomial of degree (p−e)/4, where p is an odd prime, is studied over the finite field Fp. It is shown that this factorization encodes information about the supersingular elliptic curves in Legendre normal form which admit the endomorphism , by proving an analogue of Deuring's theorem on supersingular curves with multiplier . This is used to count the number of irreducible binomial quadratic factors of P(p−e)/4(x) over Fp in terms of the class number h(−2p).