Mathematical Research Letters

Volume 22 (2015)

Number 4

On values of binary quadratic forms at integer points

Pages: 1023 – 1045



Manoj Choudhuri (Centre for Applicable Mathematics, Tata Institute of Fundamental Research, Yelahanka, Bangalore, India)

S. G. Dani (Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai (Bombay), India)


We obtain estimates for the number of integral solutions in large balls, of inequalities of the form $\lvert Q(x, y) \rvert \lt \epsilon$, where $Q$ is an indefinite binary quadratic form, in terms of the Hurwitz continued fraction expansions of the slopes of the lines on which $Q$ vanishes. The method is based on a coding of geodesics on the modular surface via Hurwitz expansions of the endpoints of their lifts in the Poincaré half-plane.

Accepted 19 September 2014

Published 24 July 2015