Ternary quadratic forms with rational zeros
Journal de théorie des nombres de Bordeaux, Tome 22 (2010) no. 1, pp. 97-113.

Formes quadratiques ternaires avec zéros rationnels

Nous considérons les formes quadratiques de Legendre

ϕ ab (x,y,z)=ax 2 +by 2 -z 2

et, en particulier, une question posée par J–P. Serre, de compter le nombre de paires d’ entiers 1aA,1bB, pour lesquels la forme ϕ ab 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.

We consider the Legendre quadratic forms

ϕ ab (x,y,z)=ax 2 +by 2 -z 2

and, in particular, a question posed by J–P. Serre, to count the number of pairs of integers 1aA,1bB, for which the form ϕ ab 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.

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
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Friedlander, John; Iwaniec, Henryk. Ternary quadratic forms with rational zeros. Journal de théorie des nombres de Bordeaux, Tome 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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