Generalization of Vélu's formulae for isogenies between elliptic curves
Miret, Josep M. (Universitat de Lleida. Departament de Matemàtica)
Moreno, Ramiro (Universitat de Lleida. Departament de Matemàtica)
Rio, Anna (Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada II)

Date:	2007
Abstract:	Given an elliptic curve E and a finite subgroup G, Vélu'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.
Rights:	Tots els drets reservats.
Language:	Anglès
Document:	Article ; recerca ; Versió publicada
Subject:	Elliptic curve ; Isogeny ; Rational subgroup
Published in:	Publicacions matemàtiques, Vol. Extra (2007) , p. 147-163, ISSN 2014-4350

Adreça alternativa: https://raco.cat/index.php/PublicacionsMatematiques/article/view/69986
DOI: 10.5565/PUBLMAT_PJTN05_07

17 p, 181.5 KB

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