Algebraic curves with many automorphisms

Let X be a (projective, geometrically irreducible, non-singular) algebraic curve of genus g defined over an algebraically closed field K of odd characteristic p. Let G be the group of all automorphisms of X which fix K element-wise. It is known that if the size of G is greater than 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 the size of G is greater than f(g) then X has zero p-rank. For even g we prove that f(g) is at most 900g^2. The odd genus case appears to be much more difficult although, for any genus g, if G has a solvable subgroup H of size larger than 252g^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 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.