Endomorphism rings of elliptic curves over finite fields by David Kohel

CHAPTER 2. ELLIPTIC CURVES AND ISOGENIES 12Proof. Silverman [29, Corollary II.2.12].Suppose that E/k is an elliptic curve over the field k. The Frobenius endomorphismrelative to k satisfies a characteristic equation π 2 −tπ +q = 0 in the ring of endomorphisms.For any extension k/k of degree r, the Frobenius endomorphism relative tok is π r . The collection of points fixed by π is exactly E(k), so the kernel of π r − 1 isE(k). Since the isogeny π −1 is separable, the cardinality of E(k) is deg(π r −1), andin particular, the number of k-rational points is deg(π − 1) = q −t+1. A theorem ofTate [31, Theorem 1] tells us that the characteristic polynomial for π determines theisogeny class of E over k.From its definition, it is clear that π commutes with all isogenies defined over k, hencewe have that π lies in the center of End k (E). The following theorem shows the keyrole that the Frobenius endmorphism plays in the structure of the elliptic curve andits endomorphism ring.Theorem 9 Let k be a perfect field of characteristic p and let E be an elliptic curveover k. Let π be the Frobenius endomorphism relative to k. The following conditionsare equivalent.1. E[p r ] = 0 for all r ≥ 1.2. The dual ̂π of the Frobenius endomorphism is purely inseparable.3. The trace of the Frobenius is divisible by p.4. The full endomorphism ring End(E) defined over an algebraic closure ofk is an order in a quaternion algebra.If the preceding equivalent conditions do not hold, then the all of the following statementshold true.1. E[p r ] = Z/p r Z for all r ≥ 1.2. The dual ̂π of the Frobenius endomorphism is separable.3. The trace of the Frobenius endomorphism is relatively prime to p.4. The endomorphism ring End(E) of E is an order in a quadratic imaginaryextension of Q.Proof. Silverman [29, Theorem V.3.1].In the first case of the theorem, we say that E is supersingular, and in the secondcase we say that E is ordinary. It is not in general true that if E is supersingularthen End k (E) is an order in a quaternion algebra.The Frobenius endomorphism determines more, however, than just these large scalestructures of the elliptic curves. The following theorem shows that the group andEnd k (E)-structure of the rational points are determined by π.Theorem 10 Let k be a finite field and let E be an elliptic curve over k, Let π be the