Endomorphism rings of elliptic curves over finite fields by David Kohel

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CHAPTER 2. ELLIPTIC CURVES AND ISOGENIES 8Associated with a quadratic space V is a quadratic map q : V → Q such thatq(u+v) −q(u) −q(v) = Φ(u, v). A quadratic module over Z is a lattice M in V suchthat the associated quadratic map on V restricts to an integer-valued map on M. Aquadratic space or quadratic module is said to be positive definite if q(v) > 0 for allnonzero v in V .Theorem 6 Let E 1 and E 2 be elliptic curves. Then there is a bilinear formΦ : Hom(E 1 , E 2 ) × Hom(E 1 , E 2 ) → Zdefined by Φ(ϕ, ψ) = ̂ϕψ+ ̂ψϕ. The bilinear form Φ defines the structure of a positivedefinite quadratic space on V = Hom(E 1 , E 2 ) ⊗ Q, with associated quadratic mapdeg, extended to V by setting deg(ϕ ⊗ r) = r 2 deg(ϕ). The lattice Hom(E 1 , E 2 ) is aquadratic module with respect to deg.Proof. [29, Corollary 6.3].As a demonstration of the quadratic module structure on Hom(E 1 , E 2 ), consider thefollowing two elliptic curves over the field k = F 41 .E 1 : y 2 = x 3 + 15x + 35E 2 : y 2 = x 3 + x + 33.The Z-module Hom(E 1 , E 2 ) is generated by isogenies ϕ and ψ of degree 3 and 7,respectively, and such thatΦ(ϕ, ψ) = ̂ϕψ + ̂ψϕ = 1.In terms of the basis {ϕ, ψ} the quadratic map deg on Hom(E 1 , E 2 ) defines a quadraticformq(x 1 , x 2 ) = deg(x 1 ϕ + x 2 ψ) = 3x 2 1 + x 1x 2 + 7x 2 2 .Such binary quadratic forms arise in the ideal theory of orders in quadratic extensionsof Q. In Chapter 3 we turn to the relation between elliptic curves and the ideal theoryof such orders. This construction of quadratic modules from isogenies of elliptic curveswill be further exploited in Chapter 6 when our principal objects of study will bequadratic modules of rank four over Z.2.2 The image of Z in End(E)We have seen that for an elliptic curve E/k, the abelian group law E × E → E isa morphism of varieties, defined over k. Silverman [29, III §2] gives explicit rationalfunctions for the maps. Thus the map[n] : EP✲ E✲ P + · · · + P
CHAPTER 2. ELLIPTIC CURVES AND ISOGENIES 9sending a point to the sum of P with itself n times is an endomorphism in End k (E),given by rational functions. From the group law on E we can recursively derive therational functions defining [n] on E. There exist relatively prime polynomials φ n , ψ n ,and ω n in k[x, y] such that [n] is given as follows.[n]E✲ E(x, y) ✲ (x n , y n ) =( φ n(x, y) ω n(x, y)ψ n (x, y) 2, ψ n (x, y) 3).Definition. The polynomials φ n , ψ n , and ω n are called the nth division polynomialson E.The polynomial ψ n plays a distinguished role in that the ideal (ψ n (x, y)) definesthe closed subscheme E[n] − {O} of E, so we may refer to ψ n as the n-th divisionpolynomial.The division polynomials satisfy many relations which can be obtained from theassociativity of the group law on E, the Weierstrass equation relating x and y, andthe explicit formulas for addition. In the case that a 1 = a 2 = a 3 = 0, Silverman [29]and Lang [17] give recursive formulas for the division polynomials. Morain [21] givesgeneral formulas for φ n and ψ n . For completeness we include here recursive formulasand relations for the division polynomials on an elliptic curve E.The division polynomial ψ n can be defined recursively via:ψ 0 = 0, ψ 1 = 1, ψ 2 = 2y + a 1 x + a 3 ,ψ 3 = 3x 4 + b 2 x 3 + 3b 4 x 2 + 3b 6 x + b 8 ,ψ 4 = ψ 2 · (2x 6 + b 2 x 5 + 5b 4 x 4 + 10b 6 x 3 + 10b 8 x 2 + (b 2 b 8 − b 4 b 6 )x + b 4 b 8 − b 2 6 )and φ n byψ 2m+1 = ψ m+2 ψ 3 m − ψ m−1 ψ 3 m+1 (m ≥ 2),ψ 2m = ψ m (ψ m+2 ψ 2 m−1 − ψ m−2ψ 2 m+1 )/ψ 2 (m > 2),φ 0 = 1, φ 1 = x, and φ n = xψ 2 n − ψ n+1ψ n−1 .Note that all of the above relations among the φ n and ψ n are generated by the relationsdefining ψ 0 , . . .,ψ 4 , φ 0 , and φ 1 , and the relations:φ r ψ 2 m − φ mψ 2 r = ψ m−rψ m+r , where r ≤ m,which can be verified directly from the group law on E. The following formula for ω nis valid if the characteristic of k is different from 2.ω n = ((ψ n+2 ψ 2 n−1 − ψ n−2 ψ 2 n+1)/ψ 2 − (a 1 φ n + a 3 ψ 2 n)ψ n )/2.This equation stems from the action of the endomorphism [n] on the invariant differential,namely thatdx dx nn =.2y + a 1 x + a 3 2y n + a 1 x n + a 3
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58Chapter 5Arithmetic of quaternion
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70Chapter 6Quadratic spacesThe equi
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94Bibliography[1] A. O. L. Atkin an
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BIBLIOGRAPHY 96[32] J. Tate. Global
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Endomorphism rings of elliptic curves over finite fields by David Kohel

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