Ternary quadratic forms with rational zeros

Friedlander, John; Iwaniec, Henryk

doi:10.5802/jtnb.706

Friedlander, John ¹; Iwaniec, Henryk ²

¹ University of Toronto 40 St. George Street Toronto, ON M5S 2E4, Canada
² Department of Mathematics Rutgers University 110 Frelinghuysen Rd. Piscataway, NJ 08903, USA

Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 1, pp. 97-113.

Abstract
Résumé

We consider the Legendre quadratic forms

ϕ_{a b} (x, y, z) = a x^{2} + b y^{2} - z^{2}

and, in particular, a question posed by J–P. Serre, to count the number of pairs of integers $1 \leq a \leq A, 1 \leq b \leq B$ , for which the form $ϕ_{a b}$ has a non-trivial rational zero. Under certain mild conditions on the integers $a, b$ , we are able to find the asymptotic formula for the number of such forms.

Formes quadratiques ternaires avec zéros rationnels

Nous considérons les formes quadratiques de Legendre

ϕ_{a b} (x, y, z) = a x^{2} + b y^{2} - z^{2}

et, en particulier, une question posée par J–P. Serre, de compter le nombre de paires d’ entiers $1 \leq a \leq A, 1 \leq b \leq B$ , pour lesquels la forme $ϕ_{a b}$ possède un zéro rationnel et non-trivial. Sous certaines conditions faibles sur les entiers $a, b$ , on peut trouver la formule asymptotique pour le nombre de telles formes.

MR Zbl | 3 citations in Numdam

DOI: 10.5802/jtnb.706

Author's affiliations:

Friedlander, John ¹; Iwaniec, Henryk ²

¹ University of Toronto 40 St. George Street Toronto, ON M5S 2E4, Canada
² Department of Mathematics Rutgers University 110 Frelinghuysen Rd. Piscataway, NJ 08903, USA

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     title = {Ternary quadratic forms with rational zeros},
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     volume = {22},
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Friedlander, John; Iwaniec, Henryk. Ternary quadratic forms with rational zeros. Journal de théorie des nombres de Bordeaux, Volume 22 (2010) no. 1, pp. 97-113. doi : 10.5802/jtnb.706. http://archive.numdam.org/articles/10.5802/jtnb.706/

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