Distorting the volcano
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Volcanoes of ℓ–isogenies of elliptic curves are a special case of graphs with a cycle called crater. In this paper, given an elliptic curve E of a volcano of ℓ–isogenies, we present a condition over an endomorphism ϕ of E in order to determine which ℓ–isogenies of E are non-descending. The endomorphism ϕ is defined as the crater cycle of an m–volcano where E is located, with m 6= ℓ. The condition is feasible when ϕ is a distortion map for a subgroup of order ℓ of E. We also provide some relationships among the crater sizes of volcanoes of m–isogenies whose elliptic curves belong to a volcano of ℓ–isogenies.
Is part ofFinite Fields and Their Applications, 2018, vol. 49, núm. C, p. 108-125
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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 ...
Miret, Josep M. (Josep Maria); Pujolàs Boix, Jordi; Valera Martín, Javier (Springer Verlag, 2015)Let E be an elliptic curve defined over a finite field Fq of odd characteristic. Let l≠2 be a prime number different from the characteristic and dividing #E(Fq). We describe how the l-adic valuation of the number of points ...
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 ...