SUPERSINGULAR ELLIPTIC CURVES AND MAXIMAL QUATERNIONIC ORDERS

We give an explicit version of the “Deuring correspondence” between supersingular elliptic curves and maximal quaternionic orders, by presenting a deterministic and explicit algorithm to compute it. In this talk all fields of positive characteristic will be either finite or function fields of curves defined over a finite algebraic extension of Fp. Elliptic curve means often isomorphy class elliptic curves. All quadratic forms have integral coefficients. For details see [4]. Let E be an elliptic curve over a finite field k of characteristic p. Such an elliptic curve is called supersingular if one, and hence all, of the following equivalences are satisfied: (1) E(k) has no p torsion; (2) Endk(E) is a 4 dimensional Z-lattice; (3) the function field k(E) has no cyclic (separable and unramified) p-extensions. Around the 30’s Helmut Hasse starts studying elliptic curves and succeeds to prove the Riemman hypothesis for zeta functions of elliptic curves. He was also the first to observe, that besides the two well known cases of endomorphism rings of elliptic curves namely Z or an order in an imaginary quadratic field extension of Q, the so called complex multiplication case -, it was also possible to have an order of a definite quaternion algebra when the base field had positive characteristic. Max Deuring a couple of years later was able to compute the discriminant of this definite algebra. In [6] he proves that the algebra ramifies at p and at ∞. Furthermore in [5] he proves that the endomorphism rings of elliptic curves over a finite field of characteristic p are maximal orders in the quaternion algebra Q∞,p (the subindex shows the ramification places of the algebra), and that all maximal orders types of that algebra appear as endomorphism rings of supersingular elliptic curves over Fp. In this talk, we want first of all, to recall more precisely this correspondence and then give an explicit and deterministic algorithm to compute it. We will show this on a concrete example. 1. Deuring correspondence In the introduction we said that Deuring proved that every maximal order type (isomorphy class) of the quaternion algebra Q∞,p appears as an endomorphism ring of a supersingular elliptic curve over Fp. But a bijection does not hold in this Date: September 24, 2004. 1