New maximal curves as ray class fields over Deligne-Lusztig curves

TL;DR

This paper constructs new maximal covers of Suzuki and Ree curves, demonstrating their maximality via ray class fields over Deligne-Lusztig curves, and establishes a connection to the Giulietti-Korchmárós curve.

Contribution

It introduces new maximal curves as ray class fields over Deligne-Lusztig curves, extending the class of known maximal curves and linking them to the Giulietti-Korchmárós curve.

Findings

Constructed new maximal covers of Suzuki and Ree curves.

Proved these covers are maximal over suitable finite fields.

Established the equivalence of the Giulietti-Korchmárós curve to a ray class field extension.

Abstract

We construct new covers of the Suzuki and Ree curves which are maximal with respect to the Hasse-Weil bound over suitable finite fields. These covers are analogues of the Giulietti-Korchm\'aros curve, which covers the Hermitian curve and is maximal over a base field extension. We show that the maximality of these curves implies that of certain ray class field extensions of each of the Deligne-Lusztig curves. Moreover, we show that the Giulietti-Korchm\'aros curve is equal to the above-mentioned ray class field extension of the Hermitian curve.