Large automorphism groups of ordinary curves in characteristic 2 (Q1734216)

An algebraic curve \(\mathcal{X}\) is said to be ordinary if its p-rank \(\gamma(\mathcal{X})\) (or Hasse-Witt invariant) equals its genus \(g(\mathcal{X})\). \textit{S. Nakajima} [Trans. Am. Math. Soc. 303, 595--607 (1987; Zbl 0644.14010)] proved that for ordinary curves of genus at least \(2\), the bound \[ |\mathrm{Aut}(\mathcal{X})|\leq 84g(\mathcal{X})(g(\mathcal{X})-1) \] holds for the order of the automorphism groups of \(\mathcal{X}\). It is an open question whether this bound is sharp or not, at least for sufficiently large \(g\) (up to the constant). Up to now, the closest known example to it is the so-called DGZ curve, see [\textit{M. Giulietti} et al., J. Number Theory 196, 114--138 (2019; Zbl 1451.14086)]. Under the assumption that \(\mathrm{Aut}(\mathcal{X})\) satisfies certain conditions, several improvements on the Nakajima's bound have been found. For instance, in odd characteristic, if \(\mathrm{Aut}(\mathcal{X})\) is solvable, in [\textit{G. Korchmáros} and \textit{M. Montanucci}, Algebra Number Theory 13, No. 1, 1--18 (2019; Zbl 1428.14048)] it is proved that \[ |\mathrm{Aut}(\mathcal{X})|\leq 34(g(\mathcal{X})+1)^{3/2}. \] In the paper under review, the authors extend this result to solvable automorphism groups of algebraic curves in characteristic \(2\), under the additional hypothesis that \(g(\mathcal{X})\) is even, proving that \[ |\mathrm{Aut}(\mathcal{X})|\leq 35(g(\mathcal{X})+1)^{3/2}. \] The proof is based on a deep analysis of certain quotient curves of \(\mathcal{X}\), exploiting the constraints on the structure of the \(2\)-Sylow subgroups of \(\mathrm{Aut}(\mathcal{X})\) given by the hypothesis of even genus.