Generalization of Vélu’s formulae for isogenies between elliptic curves
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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.
Is part ofPublicacions matemàtiques, 2007, vol. Extra, p. 147–163
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Miret, Josep M. (Josep Maria); Sadornil Renedo, Daniel; Tena Ayuso, Juan; Tomàs Cuñat, Rosa Ana; Valls Marsal, Magda (Universitat Autònoma de Barcelona. Departament de Matemàtiques, 2007)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 ...
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Fouquet, Mireille; Miret, Josep M. (Josep Maria); Valera Martín, Javier (Springer International Publishing Switzerland, 2015)Given an ordinary elliptic curve over a finite field located in the floor of its volcano of ℓ-isogenies, we present an efficient procedure to take an ascending path from the floor to the level of stability and back to ...