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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
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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 

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