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Generalization of Vélu’s formulae for isogenies between elliptic curves
Miret, Josep M. (Josep Maria); Moreno Chiral, Ramiro; Rio, Anna
Given an elliptic curve E and a finite subgroup G, V ́lu’s formulae concern to a separable isogeny IG : E → E ′ with kernel G. In particular, for a point P ∈ E these formulae express the first elementary 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 generalize 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. Simultaneously, we obtain an efficient way of performing computations concerning the isogeny when G is a rational group.
2007
Elliptic curve
Isogeny
Rational subgroup
Corbes el·líptiques
Nombres, Teoria dels
Anàlisi diofàntica
open access
(c) Universitat Autònoma de Barcelona. Departament de Matemàtiques, 2007
article
Universitat Autònoma de Barcelona. Departament de Matemàtiques
         

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