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PreprintarXiv licencearXiv

Uniqueness of the maximal function field with the third largest genus in characteristic $3$

  • Peter Beelen
  • Maria Montanucci
  • Jonathan Niemann
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Abstract

Let n be a positive integer and define q=3n. It is well known that there exists a maximal function field Z3 over 𝔽q2 of genus g3=q(qβˆ’1)/6 and that g3 is the third largest genus a maximal function field over 𝔽q2 can have. In this article we show that any maximal function field over 𝔽q2 of genus g3 is isomorphic to Z3. This extends previously known uniqueness result for the largest and second largest genera of maximal curves.

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Cite

BibTeXRIS
BibTeX
@misc{beelen2026uniqueness,
  title = {{Uniqueness of the maximal function field with the third largest genus in characteristic $3$}},
  author = {Peter Beelen and Maria Montanucci and Jonathan Niemann},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.35241},
  eprint = {2610.35241},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.35241},
}
RIS
TY  - UNPB
TI  - Uniqueness of the maximal function field with the third largest genus in characteristic $3$
AU  - Peter Beelen
AU  - Maria Montanucci
AU  - Jonathan Niemann
PY  - 2026
DA  - 2026/10/26
PB  - arXiv
DO  - 10.48550/arxiv.2610.35241
UR  - https://arxiv.org/abs/2610.35241
AB  - Let n be a positive integer and define q=3^n. It is well known that there exists a maximal function field Z_3 over 𝔽_q^2 of genus g_3=q(qβˆ’1)/6 and that g_3 is the third largest genus a maximal function field over 𝔽_q^2 can have. In this article we show that any maximal function field over 𝔽_q^2 of genus g_3 is isomorphic to Z_3. This extends previously known uniqueness result for the largest and second largest genera of maximal curves.
ER  -