On character sums with determinants

We estimate weighted character sums with determinants $ad-bc$ of $2\times 2$ matrices modulo a prime $p$ with entries $a,b,c,d$ varying over the interval $[1,N]$. Our goal is to obtain nontrivial bounds for values of $N$ as small as possible. In particular, we achieve this goal, with a power saving, for N \ge p^{1/8+\varepsilon}\ with any fixed $\epsilon >0$, which is very likely to be the best possible unless the celebrated Burgess bound is improved. By other techniques, we also treat more general sums but sometimes for larger values of $N$.