Let X be a (projective, geometrically irreducible, nonsingular) algebraic
curve of genus g≥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)∣≥8g3 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)∣≥f(g) then X has zero p-rank. For {\em{even}} g we prove
that f(g)≤900g2. The {\em{odd}} genus case appears to be much more
difficult although, for any genus g≥2, if Aut(X) has a solvable
subgroup G such that ∣G∣>252g2 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