# Generalization of Vélu's formulae for isogenies between elliptic curves.

Josep M. Miret Biosca; Ramiro Moreno; Anna Rio

Publicacions Matemàtiques (2007)

- Volume: 51, Issue: Extra, page 147-163
- ISSN: 0214-1493

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topMiret Biosca, Josep M., Moreno, Ramiro, and Rio, Anna. "Generalization of Vélu's formulae for isogenies between elliptic curves.." Publicacions Matemàtiques 51.Extra (2007): 147-163. <http://eudml.org/doc/41919>.

@article{MiretBiosca2007,

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.[Proceedings of the Primeras Jornadas de Teoría de Números (Vilanova i la Geltrú (Barcelona), 30 June - 2 July 2005)].},

author = {Miret Biosca, Josep M., Moreno, Ramiro, Rio, Anna},

journal = {Publicacions Matemàtiques},

keywords = {Teoría de números; Curvas elípticas; Geometría diofántica; elliptic curve; isogeny; rational subgroup},

language = {eng},

number = {Extra},

pages = {147-163},

title = {Generalization of Vélu's formulae for isogenies between elliptic curves.},

url = {http://eudml.org/doc/41919},

volume = {51},

year = {2007},

}

TY - JOUR

AU - Miret Biosca, Josep M.

AU - Moreno, Ramiro

AU - Rio, Anna

TI - Generalization of Vélu's formulae for isogenies between elliptic curves.

JO - Publicacions Matemàtiques

PY - 2007

VL - 51

IS - Extra

SP - 147

EP - 163

AB - 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.[Proceedings of the Primeras Jornadas de Teoría de Números (Vilanova i la Geltrú (Barcelona), 30 June - 2 July 2005)].

LA - eng

KW - Teoría de números; Curvas elípticas; Geometría diofántica; elliptic curve; isogeny; rational subgroup

UR - http://eudml.org/doc/41919

ER -