  We study a mass formula for Artin--Schreier curves of genus $g$ defined over
a finite field $k$ of characteristic $p$. For an odd prime $p$ and for small
$g$, we determine the number of $k$-isomorphism classes of Artin-Schreier
curves of genus $g$, weighted by the order of the centralizer of the
Artin-Schreier involution in the automorphism group. This extends earlier
results by several authors in characteristic $p=2$.
  Keywords: Artin-Schreier curve, finite field, automorphism, Mass formula,
moduli space, arithmetic statistics
