Hostname: page-component-848d4c4894-nr4z6 Total loading time: 0 Render date: 2024-05-04T20:27:11.649Z Has data issue: false hasContentIssue false
  • English
  • Français

SMALL SOLUTIONS OF QUADRATIC CONGRUENCES, AND CHARACTER SUMS WITH BINARY QUADRATIC FORMS

Part of: Forms and linear algebraic groups Diophantine equations Exponential sums and character sums

Published online by Cambridge University Press:  18 February 2016

D. R. Heath-Brown
D. R. Heath-Brown*
Affiliation:
Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, U.K. email rhb@maths.ox.ac.uk
Get access

Abstract

Let $Q(x,y,z)$ be an integral quadratic form with determinant coprime to some modulus $q$. We show that $q\,|\,Q$ for some non-zero integer vector $(x,y,z)$ of length $O(q^{5/8+{\it\varepsilon}})$, for any fixed ${\it\varepsilon}>0$. Without the coprimality condition on the determinant one could not necessarily achieve an exponent below $2/3$. The proof uses a bound for short character sums involving binary quadratic forms, which extends a result of Chang.

MSC classification

Primary: 11D79: Congruences in many variables
Secondary: 11L40: Estimates on character sums 11E20: General ternary and quaternary quadratic forms; forms of more than two variables
Type
Research Article
Information
Mathematika , Volume 62 , Issue 2 , 2016 , pp. 551 - 571
Copyright
Copyright © University College London 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Baker, R. C., Diophantine Inequalities (London Mathematical Society Monographs New Series 1 ) (Oxford University Press, Oxford, 1986).Google Scholar
Browning, T. D. and Heath-Brown, D. R., Rational points on quartic hypersurfaces. J. Reine Angew. Math. 629 2009, 37–88.Google Scholar
Burgess, D. A., On character sums and primitive roots. Proc. Lond. Math. Soc. (3) 12 1962, 179–192.CrossRefGoogle Scholar
Burgess, D. A., On character sums and L-series. II. Proc. Lond. Math. Soc. (3) 13 1963, 524–536.Google Scholar
Burgess, D. A., On the quadratic character of a polynomial. J. Lond. Math. Soc. 42 1967, 73–80.Google Scholar
Burgess, D. A., A note on character sums of binary quadratic forms. J. Lond. Math. Soc. 43 1968, 271–274.Google Scholar
Chang, M.-C., Burgess inequality in F p 2 . Geom. Funct. Anal. 19 2009, 1001–1016.Google Scholar
Davenport, H., Cubic forms in sixteen variables. Proc. R. Soc. Lond. Ser. A 272 1963, 285–303.Google Scholar
Gallagher, P. X. and Montgomery, H. L., On the Burgess estimate. Math. Notes 88 2010, 321–329.CrossRefGoogle Scholar
Heath-Brown, D. R., Small solutions of quadratic congruences. Glasg. Math. J. 27 1985, 87–93.Google Scholar
Heath-Brown, D. R., Small solutions of quadratic congruences, II. Mathematika 38(2) 1992, 264–284.Google Scholar
Heath-Brown, D. R., The density of rational points on cubic surfaces. Acta Arith. 79 1997, 17–30.CrossRefGoogle Scholar
Heath-Brown, D. R., Burgess’s bounds for character sums. In Number Theory and Related Fields (Springer Proceedings in Mathematics & Statistics 43 ), Springer (New York, 2013), 199–213.Google Scholar
Menchov, D., Sur les séries de fonctions orthogonales. Fund. Math. 1 1923, 82–105.CrossRefGoogle Scholar
Rademacher, H., Einige Sätze über Reihen von allgemeinen Orthogonal-Funktionen. Math. Ann. 87 1922, 112–138.CrossRefGoogle Scholar
Schinzel, A., Schlickewei, H.-P. and Schmidt, W. M., Small solutions of quadratic congruences and small fractional parts of quadratic forms. Acta Arith. 37 1980, 241–248.CrossRefGoogle Scholar