2016/12/18 by Gábor Korchmáros, Korchmáros, Gábor, Maria Montanucci +1
Mathematics · #14H05 #14H37 #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Algebraic structures and combinatorial models #FOS: Mathematics
paper · pdf · doi:10.48550/arxiv.1612.05912
openalex publication_date 2016/12/18 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
For a power q of a prime p, the Artin-Schreier-Mumford curve ASM(q) of\ngenus g=(q-1)2 is the nonsingular model \X of the irreducible\nplane curve with affine equation (Xq+X)(Yq+Y)=c, , c\≠ 0, defined over a\nfield mathbbK of characteristic p. The Artin-Schreier-Mumford curves are\nknown from the study of algebraic curves defined over a non-Archimedean\nvaluated field since for |c|<1 they are curves with a large solvable\nautomorphism group of order 2(q-1)q2 =2\√(g)(\√(g)+1)2, far away from\nthe Hurwitz bound 84(g-1) valid in zero characteristic. In this paper we deal\nwith the case where mathbbK is an algebraically closed field of\ncharacteristic p. We prove that the group Aut(\X) of all\nautomorphisms of \X fixing mathbbK elementwise has order\n2q2(q-1) and it is the semidirect product Q rtimes Dq-1 where Q is an\nelementary abelian group of order q2 and Dq-1 is a dihedral group of\norder 2(q-1). For the special case q=p, this result was proven by Valentini\nand Madan. Furthermore, we show that ASM(q) has a nonsingular model\n\Y in the three-dimensional projective space PG(3, mathbbK)\nwhich is neither classical nor Frobenius classical over the finite field\n mathbbFq2.\n