On Skabelund's Ray Class Field Covers of the Suzuki and Ree Curves

Abstract

Let $\Sm_q$ and $\Rm_q$ denote the Suzuki and Ree curves. Motivated by the Giulietti--Korchmáros curve, Skabelund constructed cyclic covers $\tSm_q$ and $\tRm_q$ of these curves and proved that they are maximal over $\F_{q^4}$ and $\F_{q^6}$, respectively. In the same paper he associated to the Suzuki and Ree curves certain ray class field covers $\Sm_{\rm rcf}$ and $\Rm_{\rm rcf}$, and showed that there are towers \[ \Sm_{\rm rcf}\longrightarrow\tSm_q\longrightarrow\Sm_q, \qquad \Rm_{\rm rcf}\longrightarrow\tRm_q\longrightarrow\Rm_q . \] Computations for small values of $q$ suggested that the first map in each tower is always an isomorphism, and the general case was left open. We prove that \[ \Sm_{\rm rcf}=\tSm_q,\qquad \Rm_{\rm rcf}=\tRm_q \] for every admissible $q$, the comparison being made over $\F_{q^4}$ in the Suzuki case and over $\F_{q^6}$ in the Ree case. The proof combines a Kummer normal form of the ray class extension, allowing a constant twist, with the centrality of its Galois group among lifted automorphisms, the standard involution of the base curve, and the first positive non-gap at the rational point at infinity.