Algebraic curves with many automorphisms

Let H be a (projective, geometrically irreducible, non-singular) algebraic curve of genus g ≥ 2 defined over an algebraically closed field K of odd characteristic p. Let Aut(H) be the group of all automorphisms of H which fix K element-wise. It is known that if |Aut(H)| ≥ 8g^3 then the p-rank (equivalently, the Hasse–Witt invariant) of H is zero. This raises the problem of determining the (minimum-value) function f(g) such that whenever |Aut(H)| ≥ f(g) then H has zero p-rank. For even g we prove that f(g) ≤ 900g^2. The odd genus case appears to be much more difficult although, for any genus g ≥ 2, if Aut(H) has a solvable subgroup G such that |G| ≥ 252g^2 then H has zero p-rank and G fixes a point of H. 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.