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

TL;DR

This paper derives explicit formulas for the $L$-polynomials of van der Geer--van der Vlugt curves over finite fields of characteristic 2, using novel methods involving Heisenberg groups and Lang torsors.

Contribution

It introduces new techniques tailored to characteristic 2 to compute $L$-polynomials of a specific class of Artin--Schreier curves, linking group characters and geometric structures.

Findings

Explicit $L$-polynomial formulas in characteristic 2

Construction of curves reaching the Hasse--Weil bound

Development of methods involving Heisenberg groups and Lang torsors

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.