Some Ree and Suzuki curves are not Galois covered by the Hermitian curve

2016/03/22 by Maria Montanucci, Montanucci, Maria, Giovanni Zini +1 · 3 citations
Mathematics · #Algebraic Geometry (math.AG) #Combinatorics (math.CO) #FOS: Mathematics #math.AG #math.CO

paper · pdf · doi:10.48550/arxiv.1603.06706

arxiv created 2016/03/22 · arxiv updated 2016/03/23

Abstract

The Deligne-Lusztig curves associated to the algebraic groups of type ²A₂, ²B₂, and ²G₂ are classical examples of maximal curves over finite fields. The Hermitian curve \mathcal H_q is maximal over \mathbb F_q², for any prime power q, the Suzuki curve \mathcal S_q is maximal over \mathbb F_q⁴, for q=2^2h+1, h≥1 and the Ree curve \mathcal R_q is maximal over \mathbb F_q⁶, for q=3^2h+1, h≥0. In this paper we show that \mathcal S₈ is not Galois covered by \mathcal H₆₄. We also give a proof for an unpublished result due to Rains and Zieve stating that \mathcal R₃ is not Galois covered by \mathcal H₂₇. Furthermore, we determine the spectrum of genera of Galois subcovers of \mathcal H₂₇, and we point out that some Galois subcovers of \mathcal R₃ are not Galois subcovers of \mathcal H₂₇.

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