  We establish an error term in the Sato-Tate theorem of Birch. That is, for
$p$ prime, $q=p^r$ we show that $\#\{ (a,b) \in \mathbb{F}_q^2 :
\theta_{a,b}\in I\} =\mu_{ST}(I)q^2 + O_r(q^{7/4})$ for any interval
$I\subseteq[0,\pi]$ where for an elliptic curve $E: y^2= x^3 +ax +b$, the
quantity $\theta_{a,b}$ is defined by $2\sqrt{q}\cos\theta_{a,b} =
q+1-E(\mathbb{F}_q)$ and $\mu_{ST}(I)$ denotes the Sato-Tate measure of the
interval $I$.

Number Theory