Ogg's Theorem via Explicit Congruences for Class Equations Ogg's Theorem via Explicit Congruences for Class Equations *

Explicit congruences (mod p) are proved for the class equations or the products of class equations corresponding to discriminants D = −8p,−3p, −12p in the theory of complex multiplication, where p is an odd prime. These congruences are used to give a new proof of a theorem of Ogg, which states that there are exactly 15 primes p for which all j-invariants of supersingular elliptic curves in characteristic p lie in the prime field Fp. The proof does not make use of any class number estimates. A corollary is that for p ≥ 13 the supersingular polynomial ssp(t) splits into linear factors (mod p) if and only if the same is true of the class equations H−8p(t), H−3p(t) (when p ≡ 1 (mod 4)) and H−12p(t).