Article ID Journal Published Year Pages File Type
5772042 Journal of Algebra 2017 22 Pages PDF
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,L2∈F‾q[T] of degree q, we construct an ordinary curve X(L1,L2) of genus g=(q−1)2 which is a generalized Artin-Schreier cover of the projective line P1. The automorphism group of X(L1,L2) over the algebraic closure F‾q of Fq contains a semidirect product Σ⋊Γ of an elementary abelian p-group Σ of order q2 by a cyclic group Γ of order q¯−1. We show that for L1≠L2, Σ⋊Γ is the full automorphism group Aut(X(L1,L2)) over F‾q; for L1=L2 there exists an extra involution and Aut(X(L1,L1))=Σ⋊Δ with a dihedral group Δ of order 2(q¯−1) containing Γ. Two different choices of the pair {L1,L2} may produce birationally isomorphic curves, even for L1=L2. We prove that any curve of genus (q−1)2 whose F‾q-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
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