Generalized Artin–Mumford curves over finite fields

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(Ljavax.xml.bind.JAXBElement@4c7e4df3,Ljavax.xml.bind.JAXBElement@2eb09190) of genus g=(q−1)2 which is a generalized Artin–Schreier cover of the projective line P1. The automorphism group of X(Ljavax.xml.bind.JAXBElement@39bd97c7,Ljavax.xml.bind.JAXBElement@fee33d5) 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(Ljavax.xml.bind.JAXBElement@3141f60e,Ljavax.xml.bind.JAXBElement@1b57cda7)) over F‾q; for L1=L2 there exists an extra involution and Aut(X(Ljavax.xml.bind.JAXBElement@4e8172a5,Ljavax.xml.bind.JAXBElement@308f903b))=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(Ljavax.xml.bind.JAXBElement@ec0e1a2,Ljavax.xml.bind.JAXBElement@2bdf3f28) 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].