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

Josep M. Miret Biosca;  Ramiro Moreno;  Anna Rio — 2007

Publicacions Matemàtiques

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...

Volcanoes of l-isogenies of elliptic curves over finite fields: The case l=3.

Josep M. Miret Biosca;  Daniel Sadornil Renedo;  Juan Tena Ayuso;  Rosana Tomàs;  Magda Valls Marsal — 2007

Publicacions Matemàtiques

This paper is devoted to the study of the volcanoes of ℓ-isogenies of elliptic curves over a finite field, focusing on their height as well as on the location of curves across its different levels. The core of the paper lies on the relationship between the ℓ-Sylow subgroup of an elliptic curve and the level of the volcano where it is placed. The particular case ℓ = 3 is studied in detail, giving an algorithm to determine the volcano of 3-isogenies of a given elliptic curve. Experimental results...

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