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SMALL SOLUTIONS OF QUADRATIC CONGRUENCES, AND CHARACTER SUMS WITH BINARY QUADRATIC FORMS
Published online by Cambridge University Press: 18 February 2016
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
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- Research Article
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- Copyright © University College London 2016
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