Explicit identities for invariants of elliptic curves

Abstract

New explicit formulas are given for the supersingular polynomial ss p ( t ) and the Hasse invariant H ˆ p ( E ) of an elliptic curve E in characteristic p. These formulas are used to derive identities for the Hasse invariants of elliptic curves E n in Tate normal form with distinguished points of order n. This yields a proof that H ˆ ( E 4 ) and H ˆ ( E 5 ) are projective invariants ( mod p ) for the octahedral group and the icosahedral group, respectively; and that the set of fourth roots λ 1 / 4 of supersingular parameters of the Legendre normal form Y 2 = X ( X − 1 ) ( X − λ ) in characteristic p has octahedral symmetry. For general n ⩾ 4 , the field of definition of a supersingular E n is determined, along with the field of definition of the points of order n on E n .