Revistes Catalanes amb Accés Obert (RACO)

Generalization of Vélu's formulae

J. M. Miret, R. Moreno, A. Rio

Resum


Given an elliptic curve $E$ and a finite subgroup $G$, Vélu's formulae concern to a separable isogeny $\mathcal{I}_G\colon E\to E'$ with kernel $G$. In particular, for a point $P\in E$ 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 $\mathcal{I}_G(P)$ 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: $w_0=t$ and $w_1=w$. We generalize this by defining parameters $w_n$ for all $n\ge 0$ 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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