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arXiv · 2610.35241

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 $q=3^n$. It is well known that there exists a maximal function field $Z_3$ over $\mathbb{F}_{q^2}$ of genus $g_3=q(q-1)/6$ and that $g_3$ is the third largest genus a maximal function field over $\mathbb{F}_{q^2}$ can have. In this article we show that any maximal function field over $\mathbb{F}_{q^2}$ 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.

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Peter Beelen, Maria Montanucci, Jonathan Niemann. 2026-10-26. Uniqueness of the maximal function field with the third largest genus in characteristic $3$. https://arxiv.org/abs/2610.35241

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