ACTA issues

## The geometry of the Artin--Schreier--Mumford curves over an algebraically closed field

Gábor Korchmáros, Maria Montanucci

Acta Sci. Math. (Szeged) 83:3-4(2017), 673-681
7/2017

 Abstract. For a power $q$ of a prime $p$, the Artin--Schreier--Mumford curve $ASM(q)$ of genus $\gg =(q-1)^2$ is the nonsingular model $\cX$ of the irreducible plane curve with affine equation $(X^q+X)(Y^q+Y)=c, c\neq0,$ defined over a field $\mathbb{K}$ of characteristic $p$. The Artin--Schreier--Mumford curves are known from the study of algebraic curves defined over a non-Archimedean valuated field since for $|c|< 1$ they are curves with a large solvable automorphism group of order $2(q-1)q^2 =2\sqrt{\gg }(\sqrt{\gg }+1)^2$, far away from the Hurwitz bound $84(\gg -1)$ valid in zero characteristic; see [Co-Ka2003-1,Co-Ka-Ko2001,Co-Ka2004]. In this paper we deal with the case where $\mathbb{K}$ is an algebraically closed field of characteristic $p$. We prove that the group $\Automorph(\cX )$ of all automorphisms of $\cX$ fixing $\mathbb{K}$ elementwise has order $2q^2(q-1)$ and it is the semidirect product $Q\rtimes D_{q-1}$ where $Q$ is an elementary abelian group of order $q^2$ and $D_{q-1}$ is a dihedral group of order $2(q-1)$. For the special case $q=p$, this result was proven by Valentini and Madan [mv1982]; see also [AK]. Furthermore, we show that $ASM(q)$ has a nonsingular model $\cY$ in the three-dimensional projective space $PG(3,\mathbb{K})$ which is neither classical nor Frobenius classical over the finite field $\mathbb{F}_{q^2}$. DOI: 10.14232/actasm-017-757-9 AMS Subject Classification (1991): 14H37, 14H05 Keyword(s): algebraic curves, algebraic function fields, positive characteristic, automorphism groups Received January 30, 2017, and in revised form September 25, 2017. (Registered under 7/2017.)