Large automorphism groups of ordinary curves of even genus in odd characteristic

Let XX be a (projective, non-singular, geometrically irreducible) curve of even genus g(X)≥2g(X)≥2 defined over an algebraically closed field KK of odd characteristic pp. If the pp-rank γ(X)γ(X) equals g(X)g(X), then XX is \emph{ordinary}. In this paper, we deal with \emph{large} automorphism groups GG of ordinary curves of even genus. We prove that |G|<821.37g(X)7/4|G|<821.37g(X)7/4. The proof of our result is based on the classification of automorphism groups of curves of even genus in positive characteristic, see \cite{giulietti-korchmaros-2017}. According to this classification, for the exceptional cases Aut(X)≅Alt7Aut(X)≅Alt7 and Aut(X)≅M11Aut(X)≅M11 we show that the classical Hurwitz bound |Aut(X)|<84(g(X)−1)|Aut(X)|<84(g(X)−1) holds, unless p=3p=3, g(X)=26g(X)=26 and Aut(X)≅M11Aut(X)≅M11; an example for the latter case being given by the modular curve X(11)X(11) in characteristic 33.