Fp2-maximal curves with many automorphisms are Galois-covered by the Hermitian curve

Daniele Bartoli; Maria Montanucci; Fernando Torres

doi:10.1515/advgeom-2021-0013

F_p2-maximal curves with many automorphisms are Galois-covered by the Hermitian curve

Daniele Bartoli^*
, Maria Montanucci
, Fernando Torres

^*Corresponding author for this work

Research output: Contribution to journal › Journal article › Research › peer-review

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Abstract

Let f be the finite field of order q². It is sometimes attributed to Serre that any curve F-covered by the Hermitian curve H_q+1 : Y^q+1 = x^q + x is also F-maximal. For prime numbers q we show that every F-maximal curve X of genus g ≤ 2 with | Aut(X) | > 84(g - 1) is Galois-covered by H_q+1. The hypothesis on | Aut(X) | is sharp, since there exists an F-maximal curve X for q = 71 of genus g = 7 with | Aut(X) | = 84(7 - 1) which is not Galois-covered by the Hermitian curve H₇₂.

Original language	English
Journal	Advances in Geometry
Volume	21
Issue number	3
Pages (from-to)	325-336
ISSN	1615-715X
DOIs	https://doi.org/10.1515/advgeom-2021-0013
Publication status	Published - 1 Jul 2021

Bibliographical note

Publisher Copyright:
© 2021 De Gruyter. All rights reserved.

Keywords

Finite field
Galois-covering
Hermitian curve
Maximal curve

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Cite this

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abstract = "Let f be the finite field of order q2. It is sometimes attributed to Serre that any curve F-covered by the Hermitian curve Hq+1 : Yq+1 = xq + x is also F-maximal. For prime numbers q we show that every F-maximal curve X of genus g ≤ 2 with | Aut(X) | > 84(g - 1) is Galois-covered by Hq+1. The hypothesis on | Aut(X) | is sharp, since there exists an F-maximal curve X for q = 71 of genus g = 7 with | Aut(X) | = 84(7 - 1) which is not Galois-covered by the Hermitian curve H72.",

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AU - Torres, Fernando

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N2 - Let f be the finite field of order q2. It is sometimes attributed to Serre that any curve F-covered by the Hermitian curve Hq+1 : Yq+1 = xq + x is also F-maximal. For prime numbers q we show that every F-maximal curve X of genus g ≤ 2 with | Aut(X) | > 84(g - 1) is Galois-covered by Hq+1. The hypothesis on | Aut(X) | is sharp, since there exists an F-maximal curve X for q = 71 of genus g = 7 with | Aut(X) | = 84(7 - 1) which is not Galois-covered by the Hermitian curve H72.

AB - Let f be the finite field of order q2. It is sometimes attributed to Serre that any curve F-covered by the Hermitian curve Hq+1 : Yq+1 = xq + x is also F-maximal. For prime numbers q we show that every F-maximal curve X of genus g ≤ 2 with | Aut(X) | > 84(g - 1) is Galois-covered by Hq+1. The hypothesis on | Aut(X) | is sharp, since there exists an F-maximal curve X for q = 71 of genus g = 7 with | Aut(X) | = 84(7 - 1) which is not Galois-covered by the Hermitian curve H72.

KW - Finite field

KW - Galois-covering

KW - Hermitian curve

KW - Maximal curve

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DO - 10.1515/advgeom-2021-0013

M3 - Journal article

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EP - 336

JO - Advances in Geometry

JF - Advances in Geometry

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