A new family of maximal curves not covered by the Hermitian curve

Abstract

For every prime power q and integer n≥1, we give a single Kummer equation defining an 𝔽_q²ⁿ-maximal curve 𝒳_q,n. The family recovers the Hermitian curves for n=1,2, the Giulietti--Korchmáros curve for n=3, and the Beelen--Montanucci curves for odd n≥5. For q>2 and even n≥4, it gives new maximal curves of genus (q²−1)qⁿ/2 that are not covered by the Hermitian curve over 𝔽_q²ⁿ. We compute the genus for both parities of n and give a uniform proof of maximality by counting rational places. For q>2 and odd n≥5, we remove the Galois hypothesis from the known non-covering results for the Beelen--Montanucci and Garcia--Güneri--Stichtenoth curves. For even n≥4, we determine the full automorphism group,of order (qⁿ+1)q(q²−1), and give explicit equations and genera for all Galois quotients, with closed-form equations for several classes of subgroups.

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BibTeX RIS

BibTeX

@misc{ma2026family,
  title = {{A new family of maximal curves not covered by the Hermitian curve}},
  author = {Liming Ma and Yipeng Wang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.19546},
  eprint = {2609.19546},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.19546},
  note = {CC BY 4.0},
}

RIS

TY  - UNPB
TI  - A new family of maximal curves not covered by the Hermitian curve
AU  - Liming Ma
AU  - Yipeng Wang
PY  - 2026
DA  - 2026/09/17
PB  - arXiv
DO  - 10.48550/arxiv.2609.19546
UR  - https://arxiv.org/abs/2609.19546
AB  - For every prime power q and integer n≥1, we give a single Kummer equation defining an 𝔽_q^2n-maximal curve 𝒳_q,n. The family recovers the Hermitian curves for n=1,2, the Giulietti--Korchmáros curve for n=3, and the Beelen--Montanucci curves for odd n≥5. For q>2 and even n≥4, it gives new maximal curves of genus (q^2−1)q^n/2 that are not covered by the Hermitian curve over 𝔽_q^2n. We compute the genus for both parities of n and give a uniform proof of maximality by counting rational places. For q>2 and odd n≥5, we remove the Galois hypothesis from the known non-covering results for the Beelen--Montanucci and Garcia--Güneri--Stichtenoth curves. For even n≥4, we determine the full automorphism group,of order (q^n+1)q(q^2−1), and give explicit equations and genera for all Galois quotients, with closed-form equations for several classes of subgroups.
ER  -

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