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

Let ${\text{Sm}}_q$ and ${\text{Rm}}_q$ denote the Suzuki and Ree curves. Motivated by the Giulietti–Korchmáros curve, Skabelund constructed cyclic covers ${\text{tSm}}_q$ and ${\text{tRm}}_q$ of these curves and proved that they are maximal over ${\text{F}}_{q^4}$ and ${\text{F}}_{q^6}$, respectively. In the same paper he associated to the Suzuki and Ree curves certain ray class field covers ${\text{Sm}}_{\mathrm{r}\mathrm{c}\mathrm{f}}$ and ${\text{Rm}}_{\mathrm{r}\mathrm{c}\mathrm{f}}$, and showed that there are towers $${\text{Sm}}_{\mathrm{r}\mathrm{c}\mathrm{f}}⟶{\text{tSm}}_q⟶{\text{Sm}}_q,{\text{Rm}}_{\mathrm{r}\mathrm{c}\mathrm{f}}⟶{\text{tRm}}_q⟶{\text{Rm}}_q\mathrm{.}$$ 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 $${\text{Sm}}_{\mathrm{r}\mathrm{c}\mathrm{f}}={\text{tSm}}_q,{\text{Rm}}_{\mathrm{r}\mathrm{c}\mathrm{f}}={\text{tRm}}_q$$ for every admissible $q$, the comparison being made over ${\text{F}}_{q^4}$ in the Suzuki case and over ${\text{F}}_{q^6}$ 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.