Endomorphism rings of elliptic curves over finite fields by David Kohel

CHAPTER 2. ELLIPTIC CURVES AND ISOGENIES 14x G and y G in the function field of E, invariant under G, as follows.x G (P) = x(P) +∑ (x(P + Q) − x(Q)),Q∈G−{0}y G (P) = y(P) +∑ (y(P + Q) − y(Q)).Q∈G−{0}(2.3)The functions x G and y G generate the function field for a curve E G and satisfy theconditions (2.1) of § 2.2 on E G . Then f G : E → E G defined by (x, y) ↦→ (x G , y G ),is an isogeny of Weierstrass equations. Under this isogeny of curves the invariantdifferential on the image curve E G pulls back to the invariant differential on E, thatis,fG ∗ ( dx Gdx) = ( ).2y G + a 1 x G + a 3 2y + a 1 x + a 3Following Vélu [33], we can write down explicit equations for x G and y G in terms ofx and y defining the isogeny f G of curves with the kernel specified by ψ(x) in k[x].He develops rational functions in terms of the roots of ψ(x), but the isogeny is moreappropriately expressed in terms of symmetric functions in the roots as follows.Isogenies of odd degreeFirst we assume that the degree of the isogeny determined by the equation ψ(x) forthe kernel is odd. A general isogeny over k can be decomposed over k into a compositeof isogenies of degree 2 or 4 and isogenies of odd degree. We will treat decompositionof G in the sequel.The isogeny is described in terms of the coefficients of ψ(x) as follows.where φ(x) is given by(x, y) ↦−→ (x G , y G ) = ( φ(x) ω(x, y)ψ(x) 2, ψ(x) ), 3φ(x) = (4x 3 + b 2 x 2 + 2b 4 x + b 6 )(ψ ′ (x) 2 − ψ ′′ (x)ψ(x))−(6x 2 + b 2 x + b 4 )ψ ′ (x)ψ(x) + (dx − 2s 1 )ψ(x) 2 ,where the degree of the isogeny is d = 2n+1, and s i is the ith elementary symmetricfunction in the roots of ψ(x), so that ψ(x) = x n − s 1 x n−1 + · · · + (−1) n s n .If the characteristic of the base field k is different from 2, one can derive the equationfor ω(x, y) from φ(x) and ψ(x) using the condition that the the invariant differentialon E G pulls back to the invariant differential on E.ω(x, y) = φ ′ (x)ψ(x)ψ 2 /2 − φ(x)ψ ′ (x)ψ 2 + (a 1 φ(x) + a 3 ψ(x) 2 ))ψ(x)/2.