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

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.

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.

Reçu le :
Publié le :
DOI : https://doi.org/10.5802/jtnb.706
@article{JTNB_2010__22_1_97_0,
     author = {John Friedlander and Henryk Iwaniec},
     title = {Ternary quadratic forms with rational zeros},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {97--113},
     publisher = {Universit\'e Bordeaux 1},
     volume = {22},
     number = {1},
     year = {2010},
     doi = {10.5802/jtnb.706},
     zbl = {1219.11060},
     mrnumber = {2675875},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.706/}
}
John Friedlander; Henryk Iwaniec. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.706/

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