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Denis Simon
Sur la paramétrisation des solutions des équations quadratiques
Journal de théorie des nombres de Bordeaux, 18 no. 1 (2006), p. 265-283, doi: 10.5802/jtnb.543
Article PDF | Reviews MR 2245885 | Zbl 05070457
Keywords: Formes quadratiques binaires et ternaires, paramétrisation, groupe de classes

Résumé - Abstract

Our goal in this paper is to give a link between different classical aspects of the theory of integral quadratic forms. First, we investigate the properties of the binary quadratic forms involved in the parametrization of the solutions of ternary quadratic equations. In particular, we exhibit a simple rule to obtain a parametrization from a particular solution, such that its invariants only depend on the original equation. Used in the context of elliptic curves, this parametrization simplifies the algorithm of $2$-descent.

Secondly, we consider a primitive quadratic form $Q(X,Y)$, with nonsquare discriminant. Some authors (in [1] and [7]) make a link between a particular rational solution of $Q(X,Y)=1$ over $\mathbb{Q}^2$ and a solution of $[R]^2=[Q]$ in the class group $Cl(\Delta )$. We explain why this link is much more direct than this. Indeed, when the equation $Q(X,Y)=1$ has a solution, it is possible to parametrize them all by $ X=\frac{q_1(s,t)}{q_3(s,t)}$ and $ Y=\frac{q_2(s,t)}{q_3(s,t)}$ where $q_1$,$q_2$ and $q_3$ are three integral quadratic forms with $\operatorname{Disc} q_3 = \Delta $. We show that the quadratic form $q_3$ is exactly (up to sign) the solution $R$ of $[R]^2=[Q]$ in $Cl(\Delta )$. We end by a comparison between our algorithm for extracting square roots of quadratic forms and the algorithm of Gauss.

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