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Title:
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Generalization of Vélu’s formulae for isogenies between elliptic curves
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Author:
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Miret, Josep M. (Josep Maria); Moreno Chiral, Ramiro; Rio, Anna
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Notes:
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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. |
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Subject(s):
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-Elliptic curve
-Isogeny
-Rational subgroup
-Corbes el·líptiques
-Nombres, Teoria dels
-Anàlisi diofàntica |
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Rights:
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(c) Universitat Autònoma de Barcelona. Departament de Matemàtiques, 2007
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Document type:
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article
publishedVersion |
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Published by:
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Universitat Autònoma de Barcelona. Departament de Matemàtiques
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