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

Journal article

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.

Authors: Heath-Brown, R

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: Cambridge University Press
• Journal: Mathematika
• Volume: 62
• Issue: 2
• Pages: 551-571
• Publication date: 2016-02-18
• DOI: 10.1112/S0025579315000364
• EISSN: 2041-7942
• ISSN: 0025-5793
• Language: English