A Hecke endomorphism algebra is a natural generalisation of the qSchur algebra associated with the symmetric group to a Coxeter group. For Weyl groups, Parshall, Scott and the first author [9, 10] investigated the stratification structure of these algebras in order to seek applications to representations of finite groups of Lie type. In this paper we investigate the presentation problem for Hec...

Theorem 15.1. Let E/Fq be an elliptic curve over a finite field, and let πE be the Frobenius endomorphism of E. Then E is supersingular if and only if trπE ≡ 0 mod p. Proof. Let q = pn and let π be the p-power Frobenius map π(x, y) = (xp, yp) (note that π is an isogeny, but not necessarily an endomorphism, since E need not be defined over Fp). We have π̂π = [p], where [p] denotes the multiplicat...