here - Sites personnels de TELECOM ParisTech - Télécom ParisTech

104 Chapter 3. Bent functions and algebraic curves3.2.2 Elliptic curves over finite fieldsWe now focus on elliptic curves defined over finite fields. Let m be a positive integer, F q thefinite field of characteristic p with q = p m elements and F q its algebraic closure.The group of rational points of E over an extension F q k of F q (i.e. points with coordinates inF q k) is denoted by E(F q k); the number of points of this group by #E(F q k). When the context isclear, we denote #E(F q ) simply by #E.Definition 3.2.8 (Frobenius endomorphism). Let E be an elliptic curve defined over the finitefield F q of characteristic p. Then the q-th power map is an endomorphism of E called the q-thFrobenius endomorphism of the curve.It is a classical result that #E = q + 1 − t where t is the trace of the Frobenius endomorphismof E over F q and the following theorem has been shown by Hasse.Theorem 3.2.9 ([244, Theorem V.2.3.1]). Let t be the trace of the Frobenius endomorphism ofan elliptic curve defined over F q , then|t| ≤ 2 √ q .Here we will be interested in ordinary elliptic curves which can be defined as follows.Definition 3.2.10 (Ordinary elliptic curve [244, Theorem V.3.1]). Let E be an elliptic curvedefined over F q of characteristic p and t the trace of the Frobenius endomorphism of E. We saythat E is ordinary if it verifies one of the following equivalent properties:• p ∤ t;• E[p] ≃ Z/pZ;• End(E) is an order in an imaginary quadratic extension of Q.If E is not ordinary, we say it is supersingular.Finally, using classical results of Deuring [68] and Waterhouse [280], the number of ordinaryelliptic curves (up to isomorphism) with a given trace t of the Frobenius endomorphism (orequivalently a number of points q + 1 − t), verifying |t| ≤ 2 √ q and p ∤ t, can be computed asfollows 11 . The conditions on t indeed imply that End(E) must be an order O in K = Q[α] andcontains the order Z[α] of discriminant ∆ where α = t+√ ∆2and ∆ = t 2 − 4q. We denote by H(∆)the Kronecker class number [232, 61]H(∆) =∑Z[α]⊂O⊂Kh(O) ,where the sum is taken over all the orders O in K containing Z[α] and h(O) is the classical classnumber.Proposition 3.2.11 ([232, 144, 61]). Let t be an integer such that |t| ≤ 2 √ q and p ∤ t. Thenumber N(t) of elliptic curves over F q with q + 1 − t rational points is given bywhere ∆ = t 2 − 4q.11 See Chapter 5 for details.N(t) = H(∆) ,