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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(q5/8+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:
Published
Peer review status:
Peer reviewed

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Publisher copy:
10.1112/S0025579315000364

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Institution:
University of Oxford
Division:
MPLS
Department:
Mathematical Institute
Role:
Author
Publisher:
Cambridge University Press Publisher's website
Journal:
Mathematika Journal website
Volume:
62
Issue:
2
Pages:
551-571
Publication date:
2016-02-18
DOI:
EISSN:
2041-7942
ISSN:
0025-5793
Source identifiers:
572240
Language:
English
Pubs id:
pubs:572240
UUID:
uuid:1813b94b-cfd6-4a3b-8148-59433f1df5ea
Local pid:
pubs:572240
Deposit date:
2015-11-04

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