Generalization of Vélu’s formulae for isogenies between elliptic curves

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2007
Author
Miret, Josep M. (Josep Maria)
Moreno Chiral, Ramiro
Rio, Anna
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Miret, Josep M. (Josep Maria); Moreno Chiral, Ramiro; Rio, Anna; . (2007) . Generalization of Vélu’s formulae for isogenies between elliptic curves. Publicacions matemàtiques, 2007, vol. Extra, p. 147–163. https://doi.org/10.5565/PUBLMAT_PJTN05_07.
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Abstract
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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http://hdl.handle.net/10459.1/44519
DOI
https://doi.org/10.5565/PUBLMAT_PJTN05_07
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Publicacions matemàtiques, 2007, vol. Extra, p. 147–163
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