Ternary quadratic forms with rational zeros
Journal de Théorie des Nombres de Bordeaux, Volume 22 (2010) no. 1, pp. 97-113.

We consider the Legendre quadratic forms

ϕab(x,y,z)=ax2+by2-z2

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.

Formes quadratiques ternaires avec zéros rationnels

Nous considérons les formes quadratiques de Legendre

ϕab(x,y,z)=ax2+by2-z2

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.

Received:
Published online:
DOI: 10.5802/jtnb.706
John Friedlander 1; Henryk Iwaniec 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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John Friedlander; Henryk Iwaniec. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.706/

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