Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5772042 | Journal of Algebra | 2017 | 22 Pages |
Abstract
Let Fq be the finite field of order q=ph with p>2 prime and h>1, and let Fq¯ be a subfield of Fq. From any two q¯-linearized polynomials L1,L2âFâ¾q[T] of degree q, we construct an ordinary curve X(L1,L2) of genus g=(qâ1)2 which is a generalized Artin-Schreier cover of the projective line P1. The automorphism group of X(L1,L2) over the algebraic closure Fâ¾q of Fq contains a semidirect product ΣâÎ of an elementary abelian p-group Σ of order q2 by a cyclic group Î of order q¯â1. We show that for L1â L2, ΣâÎ is the full automorphism group Aut(X(L1,L2)) over Fâ¾q; for L1=L2 there exists an extra involution and Aut(X(L1,L1))=ΣâÎ with a dihedral group Î of order 2(q¯â1) containing Î. Two different choices of the pair {L1,L2} may produce birationally isomorphic curves, even for L1=L2. We prove that any curve of genus (qâ1)2 whose Fâ¾q-automorphism group contains an elementary abelian subgroup of order q2 is birationally equivalent to X(L1,L2) for some separable q¯-linearized polynomials L1,L2 of degree q. We produce an analogous characterization in the special case L1=L2. This extends a result on the Artin-Mumford curves, due to Arakelian and Korchmáros [1].
Related Topics
Physical Sciences and Engineering
Mathematics
Algebra and Number Theory
Authors
Maria Montanucci, Giovanni Zini,