On maximal curves that are not quotients of the Hermitian curve

2015/11/17 by Massimo Giulietti, Giulietti, Massimo, Maria Montanucci +3 · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research

paper · pdf · doi:10.48550/arxiv.1511.05353

openalex publication_date 2015/11/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

For each prime power ℓ the plane curve \mathcal X_ℓ with equation Y^ℓ2-ℓ+1=X^ℓ2-X is maximal over \mathbbF_ℓ⁶. Garcia and Stichtenoth in 2006 proved that \mathcal X₃ is not Galois covered by the Hermitian curve and raised the same question for \mathcal X_ℓ with ℓ>3; in this paper we show that \mathcal X_ℓ is not Galois covered by the Hermitian curve for any ℓ>3. Analogously, Duursma and Mak proved that the generalized GK curve \mathcal C_ℓⁿ over \mathbbF_ℓ²ⁿ is not a quotient of the Hermitian curve for ℓ>2 and n≥ 5, leaving the case ℓ=2 open; here we show that \mathcal C_2ⁿ is not Galois covered by the Hermitian curve over \mathbbF_2²ⁿ for n≥5.

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• Non-isomorphic subfields of the BM and GGS maximal function fields

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