On values of isotropic quadratic forms

Abstract

Let $K$ be a locally compact non-discrete field of characteristic $p>2$ and $Q$ be a non-degenerate isotropic binary quadratic form with coefficients in $K$. We obtain asymptotic estimates for the number of solutions in the two-fold product of a discrete subring inside $K$, of the inequalities of the form $|Q(x,y)|<\delta$ for some $\delta>0$, where $| \cdot |$ is an ultrametric absolute value on $K$. The estimates are obtained in terms of continued fraction expansions of the coefficients of the quadratic form $Q$.