2016/12/06 by Maria Montanucci, Montanucci, Maria, Giovanni Zini +1
Computer Science · Mathematics · #14H05 #14H37 #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Coding theory and cryptography #FOS: Mathematics #Finite Group Theory Research
paper · pdf · doi:10.48550/arxiv.1612.01731
openalex publication_date 2016/12/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let \mathbbFq be the finite field of order q=ph with p>2 prime and h>1, and let \mathbbF_q be a subfield of \mathbbFq. From any two q-linearized polynomials L1,L2 ∈ \mathbbFq[T] of degree q, we construct an ordinary curve X(L1,L2) of genus (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 \mathbbFq of \mathbbFq contains a semidirect product Σ\rtimes 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\rtimes Γ is the full automorphism group \rm Aut(X(L1,L2)) over \mathbbFq; for L1=L2 there exists an extra involution and \rm Aut(X(L1,L1))=Σ\rtimes 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 \mathbbFq-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.