Ternary quadratic forms with rational zeros

John Friedlander; Henryk Iwaniec

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

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

DOI: 10.5802/jtnb.706

Author's affiliations:

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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     volume = {22},
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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/

References
Cited by

[1] Cojocaru A. C. and Murty M. R., An Introduction to Sieve Methods and their Applications. London Math. Soc. Student Texts 66. Cambridge University Press, Cambridge, 2005. | MR | Zbl

[2] Guo C. R., On solvability of ternary quadratic forms. Proc. London Math. Soc. 70 (1995), 241–263. | MR | Zbl

[3] Fouvry É. and Klüners J., On the 4-rank of class groups of quadratic number fields. Invent. Math. 167 (2007), 455–513. | MR | Zbl

[4] Heilbronn H., On the averages of some arithmetical functions of two variables. Mathematika 5 (1958), 1–7. | MR | Zbl

[5] Iwaniec H., Rosser’s sieve. Acta Arith. 36 (1980), 171–202. | MR | Zbl

[6] Serre J–P., A Course of Arithmetic. Springer, New York, 1973. | Zbl

[7] Serre J–P., Spécialisation des éléments de ${Br}_{2} (Q (T_{1}, \dots, T_{n}))$ . C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), 397–402. | MR | Zbl

[8] Titchmarsh E. C., The Theory of the Riemann Zeta-Function, 2nd ed., revised by D.R. Heath-Brown. Clarendon Press, Oxford, 1986. | MR | Zbl

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