Canonical Projective Embeddings of the Deligne-Lusztig Curves Associated to $2A2$, $2B2$ and $2G2$

TL;DR

This paper constructs explicit projective models for Deligne-Lusztig curves associated with certain algebraic groups, providing polynomial equations that describe their rational points over various finite fields.

Contribution

It introduces explicit projective embeddings of these curves and identifies polynomials characterizing their rational points over different finite field extensions.

Findings

Explicit projective models of the curves are constructed.

Polynomials for rational points over finite fields are identified.

Relations among these polynomials are demonstrated.

Abstract

The Deligne-Lusztig varieties associated to the Coxeter classes of the algebraic groups 2A2, 2B2 and 2G2 are affine algebraic curves. We produce explicit projective models of the closures of these curves. Furthermore for $d$ the Coxeter number of these groups, we find polynomials for each of these models that cut out the $\F_q$-points, the $\F_{q^d}$-points and the $\F_{q^{d+1}}$-points, and demonstrate a relation satisfied by these polynomials.