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3.2. Algebraic curves 103Proposition 3.2.3 ([244, Theorem III.4.8]). Let E 1 and E 2 be two elliptic curves defined overK and φ a map between them. If φ is an isogeny, then φ is a homomorphism.We denote by End K (E) the ring of rational endomorphisms (i.e. homomorphisms or equivalentlyisogenies from E to itself) and by End(E) = End K(E) the ring of endomorphisms of Eover the algebraic closure K of K. It is possible to define the degree [244, Section II.2] and thetrace [244, Proposition III.8.6] of such maps. Before stating the theorem giving the structure ofEnd K (E), we need different definitions.Definition 3.2.4 (Order [244, Section III.9]). Let K be a Q-algebra finitely generated over Q.An order O in K is a subring of K which is finitely generated and such that 7 O ⊗ Z Q = K.In a number field K of degree [K : Q] = n, an order O is a subring which is also a lattice, i.e.a Z-module of rank n 8 .Definition 3.2.5 (Definite quaternion algebra [244, Section III.9]). A definite quaternion algebraK over Q is an algebra of the formwhose multiplication satisfiesK = Q + Qα + Qβ + Qαβ ,α 2 , β 2 ∈ Q, α 2 , β 2 < 0, αβ = −βα .Theorem 3.2.6 ([244, Corollary III.9.4]). The endomorphism ring of an elliptic curve E overK is one of the three following objects:1. the ring of natural integers Z;2. an order in an imaginary quadratic number field 9 ;3. a maximal order in a definite quaternion algebra.Moreover, if char(K) = 0, then only the first two are possible.In particular, End(E) is torsion-free. If End(E) is strictly larger than Z, then the curve issaid to have complex multiplication 10 .The group of rational points of E over an extension L of K (i.e. points with coordinates in L)is denoted by E(L). For an integer n, we denote by E[n] the n-torsion subgroup of the points ofE over K, i.e.E[n] = { P ∈ E(K) | [n]P = O E}.The subgroup of rational points of n-torsion is denoted by E[n](K) = E[n] ∩ E(K). The followingclassical result gives the structure of the groups of torsion points.Proposition 3.2.7 ([244, Corollary III.6.4]). Let n be a positive integer and p the characteristicof K. Then:• If p ≠ 0 and p ∤ n, then E[n] ≃ Z/nZ × Z/nZ.• One of the following is true: E[p e ] ≃ {0} for all e ≥ 1 or E[p e ] ≃ Z/p e Z for all e ≥ 1.It can also be shown that a point of E is of n-torsion if and only if its coordinates are roots ofa bivariate polynomial called the n-division polynomial of E [18, Section III.4]. In fact, one caneven choose a univariate polynomial in the x-coordinate and we denote that polynomial by f n .7 The operator ⊗ Z denotes a tensor product of Z-modules.8 More details about such orders will be given in Chapters 5 and6.9 An imaginary quadratic number field is a number field of degree 2 whose complex embeddings are complex,i.e. their images are not included in the real numbers R.10 This definition is only valid in dimension 1. For general abelian varieties, the “right” definition is given inChapter 6.