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Small solutions of quadratic congruences, and character sums with binary quadratic forms

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+ε ), for any fixed ε > 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.
Publication status:
Accepted
Peer review status:
Peer reviewed
Version:
Accepted Manuscript

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Institution:
University of Oxford
Department:
Oxford, MPLS, Mathematical Institute
Role:
Author
Publisher:
University College London, Faculty of Mathematical and Physical Sciences, Department of Mathematics Publisher's website
Journal:
Mathematika: a journal of pure and applied mathematics Journal website
Publication date:
2015-01-01
EISSN:
2041-7942
ISSN:
0025-5793
URN:
uuid:1813b94b-cfd6-4a3b-8148-59433f1df5ea
Source identifiers:
572240
Local pid:
pubs:572240

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