Generalization of Vélu’s formulae for isogenies between elliptic curves
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 t...
| Autores: | , , |
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| Formato: | artículo |
| Estado: | Versión publicada |
| Fecha de publicación: | 2007 |
| País: | España |
| Recursos: | Universitat de Lleida (UdL) |
| Repositorio: | Repositori Obert UdL |
| OAI Identifier: | oai:repositori.udl.cat:10459.1/44519 |
| Acesso em linha: | https://doi.org/10.5565/PUBLMAT_PJTN05_07 http://hdl.handle.net/10459.1/44519 |
| Access Level: | acceso abierto |
| Palavra-chave: | Elliptic curve Isogeny Rational subgroup Corbes el·líptiques Nombres, Teoria dels Anàlisi diofàntica |
| Resumo: | 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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