Publ. Mat. (2007), 147–163 Proceedings of the Primeras Jornadas de Teoŕıa de Números. GENERALIZATION OF VÉLU’S FORMULAE FOR ISOGENIES BETWEEN ELLIPTIC CURVES

Given an elliptic curve E and a finite subgroup G, Vélu’s formu-lae concern to a separable isogeny IG: E → E ′ with kernel G. In particular, for a point P ∈ E these formulae express the first el-ementary symmetric polynomial on the abscissas of the points in the set P +G as the difference between the abscissa of IG(P) and the first elementary symmetric polynomial on the abscissas of the nontrivial points of the kernel G. On the other hand, they express Weierstraß coefficients of E ′ as polynomials in the coefficients of E and two additional parameters: w0 = t and w1 = w. We gener-alize this by defining parameters wn for all n ≥ 0 and giving analogous formulae for all the elementary symmetric polynomials and the power sums on the abscissas of the points in P+G. Simul-taneously, we obtain an efficient way of performing computations concerning the isogeny when G is a rational group. 1. Vélu’s formulae An elliptic curve E over a field K is a smooth projective curve over K of genus one with a distinguished K-rational point. If E/K is an elliptic curve then it has a plane model given by an affine Weierstraß equation y2 + a1xy + a3y = x 3 + a2x 2 + a4x+ a6 such that the coefficients ai ∈ K and the distinguished point is the point at infinity O = (0: 1: 0) ∈ E(K)