On the pinned distances problem in positive characteristic

We study the Erdos-Falconer distance problem for a set A subset of F2$A\subset \mathbb {F}<^>2$, where F$\mathbb {F}$ is a field of positive characteristic p$p$. If F=Fp$\mathbb {F}=\mathbb {F}_p$ and the cardinality |A|$|A|$ exceeds p5/4$p<^>{5/4}$, we prove that A$A$ determines an asymptotically full proportion of the feasible p$p$ distances. For small sets A$A$, namely when |A|<= p4/3$|A|\leqslant p<^>{4/3}$ over any F$\mathbb {F}$, we prove that either A$A$ determines >>|A|2/3$\gg |A|<^>{2/3}$ distances, or A$A$ lies on an isotropic line. For both large and small sets, the results proved are in fact for pinned distances.