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

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.