Formes quadratiques ternaires avec zéros rationnels
Nous considérons les formes quadratiques de Legendre
et, en particulier, une question posée par J–P. Serre, de compter le nombre de paires d’ entiers , pour lesquels la forme possède un zéro rationnel et non-trivial. Sous certaines conditions faibles sur les entiers , on peut trouver la formule asymptotique pour le nombre de telles formes.
We consider the Legendre quadratic forms
and, in particular, a question posed by J–P. Serre, to count the number of pairs of integers , for which the form has a non-trivial rational zero. Under certain mild conditions on the integers , we are able to find the asymptotic formula for the number of such forms.
@article{JTNB_2010__22_1_97_0,
author = {Friedlander, John and Iwaniec, Henryk},
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 = {http://www.numdam.org/articles/10.5802/jtnb.706/}
}
TY - JOUR AU - Friedlander, John AU - Iwaniec, Henryk TI - Ternary quadratic forms with rational zeros JO - Journal de théorie des nombres de Bordeaux PY - 2010 SP - 97 EP - 113 VL - 22 IS - 1 PB - Université Bordeaux 1 UR - http://www.numdam.org/articles/10.5802/jtnb.706/ DO - 10.5802/jtnb.706 LA - en ID - JTNB_2010__22_1_97_0 ER -
%0 Journal Article %A Friedlander, John %A Iwaniec, Henryk %T Ternary quadratic forms with rational zeros %J Journal de théorie des nombres de Bordeaux %D 2010 %P 97-113 %V 22 %N 1 %I Université Bordeaux 1 %U http://www.numdam.org/articles/10.5802/jtnb.706/ %R 10.5802/jtnb.706 %G en %F JTNB_2010__22_1_97_0
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://www.numdam.org/articles/10.5802/jtnb.706/
[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 . 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
Cité par Sources :
