Sur la paramétrisation des solutions des équations quadratiques
Denis Simon
Journal de Théorie des Nombres de Bordeaux, Volume 18 (2006) no. 1, p. 265-283

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 2 and a solution of [R] 2 =[Q] in the class group Cl(Δ). 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=q 1 (s,t) q 3 (s,t) and Y=q 2 (s,t) q 3 (s,t) where q 1 ,q 2 and q 3 are three integral quadratic forms with Discq 3 =Δ. We show that the quadratic form q 3 is exactly (up to sign) the solution R of [R] 2 =[Q] in Cl(Δ). We end by a comparison between our algorithm for extracting square roots of quadratic forms and the algorithm of Gauss.

L’objectif de cet article est de proposer un lien entre plusieurs aspects classiques de la théorie des formes quadratiques entières. Dans un premier temps, on étudie en détail les propriétés des formes quadratiques binaires qui paramétrisent les solutions des équations quadratiques ternaires. En particulier, on donne un moyen simple de construire une paramétrisation à partir d’une solution particulière, dont les invariants ne dépendent que de l’équation de départ. Cette paramétrisation permet de simplifier l’algorithme de la 2-descente sur les courbes elliptiques.

Dans un deuxième temps, on considère Q(X,Y) une forme quadratique entière primitive de discriminant Δ non carré. Certains auteurs (dans [1] et [7]) dressent un lien entre une solution rationnelle particulière de Q(X,Y)=1 dans 2 et une solution de [R] 2 =[Q] dans le groupe de classes Cl(Δ). Nous montrons que ce lien est bien plus direct que celui décrit dans [1] et [7]. En effet, lorsque l’équation Q(X,Y)=1 admet une solution, il est possible de paramétrer toutes les solutions sous la forme X=q 1 (s,t) q 3 (s,t) et Y=q 2 (s,t) q 3 (s,t)q 1 ,q 2 et q 3 sont trois formes quadratiques entières avec Discq 3 =Δ. Nous montrons que la forme quadratique q 3 est exactement (au signe près) la solution R de l’équation [R] 2 =[Q] dans Cl(Δ). Nous comparons alors notre algorithme d’extraction de racine carrée de forme quadratique, avec celui de Gauss.

Received : 2004-03-29
Published online : 2008-12-03
DOI : https://doi.org/10.5802/jtnb.543
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     author = {Denis Simon},
     title = {Sur la param\'etrisation des solutions des \'equations quadratiques},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {18},
     number = {1},
     year = {2006},
     pages = {265-283},
     doi = {10.5802/jtnb.543},
     zbl = {05070457},
     mrnumber = {2245885},
     language = {fr},
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Simon, Denis. Sur la paramétrisation des solutions des équations quadratiques. Journal de Théorie des Nombres de Bordeaux, Volume 18 (2006) no. 1, pp. 265-283. doi : 10.5802/jtnb.543. jtnb.centre-mersenne.org/item/JTNB_2006__18_1_265_0/

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