  Let $X$ be a (projective, geometrically irreducible, nonsingular) algebraic
curve of genus $g \ge 2$ defined over an algebraically closed field $K$ of odd
characteristic $p$. Let $Aut(X)$ be the group of all automorphisms of $X$ which
fix $K$ element-wise. It is known that if $|Aut(X)|\geq 8g^3$ then the $p$-rank
(equivalently, the Hasse-Witt invariant) of $X$ is zero. This raises the
problem of determining the (minimum-value) function $f(g)$ such that whenever
$|Aut(X)|\geq f(g)$ then $X$ has zero $p$-rank. For {\em{even}} $g$ we prove
that $f(g)\leq 900 g^2$. The {\em{odd}} genus case appears to be much more
difficult although, for any genus $g\geq 2$, if $Aut(X)$ has a solvable
subgroup $G$ such that $|G|>252 g^2$ then $X$ has zero $p$-rank and $G$ fixes a
point of $X$. Our proofs use the Hurwitz genus formula and the Deuring
Shafarevich formula together with a few deep results from finite group theory
characterizing finite simple groups whose Sylow $2$-subgroups have a cyclic
subgroup of index $2$. We also point out some connections with the Abhyankar
conjecture and the Katz-Gabber covers.
