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.