Ternary quadratic forms with rational zeros

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 (2010)

  • Volume: 22, Issue: 1, page 97-113
  • ISSN: 1246-7405

Abstract

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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 ≤ a ≤ A , 1 ≤ b ≤ 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.

How to cite

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Friedlander, John, and Iwaniec, Henryk. "Ternary quadratic forms with rational zeros." Journal de Théorie des Nombres de Bordeaux 22.1 (2010): 97-113. <http://eudml.org/doc/116402>.

@article{Friedlander2010,
abstract = {We consider the Legendre quadratic forms\[\varphi \_\{ab\}(x,y,z)= ax^2+by^2-z^2 \]and, in particular, a question posed by J–P. Serre, to count the number of pairs of integers $1\le a\le A, \, 1\le b \le B$, for which the form $\varphi _\{ab\}$ 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.},
affiliation = {University of Toronto 40 St. George Street Toronto, ON M5S 2E4, Canada; Department of Mathematics Rutgers University 110 Frelinghuysen Rd. Piscataway, NJ 08903, USA},
author = {Friedlander, John, Iwaniec, Henryk},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {quadratic forms; non-trivial zero; character sums with multiplicative coefficients},
language = {eng},
number = {1},
pages = {97-113},
publisher = {Université Bordeaux 1},
title = {Ternary quadratic forms with rational zeros},
url = {http://eudml.org/doc/116402},
volume = {22},
year = {2010},
}

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
PB - Université Bordeaux 1
VL - 22
IS - 1
SP - 97
EP - 113
AB - We consider the Legendre quadratic forms\[\varphi _{ab}(x,y,z)= ax^2+by^2-z^2 \]and, in particular, a question posed by J–P. Serre, to count the number of pairs of integers $1\le a\le A, \, 1\le b \le B$, for which the form $\varphi _{ab}$ 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.
LA - eng
KW - quadratic forms; non-trivial zero; character sums with multiplicative coefficients
UR - http://eudml.org/doc/116402
ER -

References

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  4. Heilbronn H., On the averages of some arithmetical functions of two variables. Mathematika 5 (1958), 1–7. Zbl0125.02604MR97362
  5. Iwaniec H., Rosser’s sieve. Acta Arith. 36 (1980), 171–202. Zbl0435.10029MR581917
  6. Serre J–P., A Course of Arithmetic. Springer, New York, 1973. Zbl0256.12001
  7. Serre J–P., Spécialisation des éléments de Br 2 ( Q ( T 1 , ⋯ , T n ) ) . C. R. Acad. Sci. Paris Sér. I Math. 311 (1990), 397–402. Zbl0711.13002MR1075658
  8. Titchmarsh E. C., The Theory of the Riemann Zeta-Function, 2nd ed., revised by D.R. Heath-Brown. Clarendon Press, Oxford, 1986. Zbl0601.10026MR882550

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