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arXiv · 2505.22036

The $L$-polynomials of van der Geer--van der Vlugt curves in characteristic $2$

Tetsushi Ito·Daichi Takeuchi·Takahiro Tsushima
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Abstract

The van der Geer--van der Vlugt curves form a class of Artin--Schreier coverings of the projective line over finite fields. We provide an explicit formula for their $L$-polynomials in characteristic $2$, expressed in terms of characters of maximal abelian subgroups of associated Heisenberg groups. For this purpose, we develop new methods specific to characteristic $2$ that exploit the structure of the Heisenberg groups and the geometry of Lang torsors for $W_2$. As an application, we construct examples of curves in this family attaining the Hasse--Weil bound.

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Tetsushi Ito, Daichi Takeuchi, Takahiro Tsushima. 2025-05-28. The $L$-polynomials of van der Geer--van der Vlugt curves in characteristic $2$. https://arxiv.org/abs/2505.22036

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