发表机构
Universidade Estadual de Campinas（坎皮纳斯州立大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文证明了Skabelund构造的Suzuki与Ree曲线的射线类域覆盖分别等于其循环覆盖，对所有可容许的q成立，结合Kummer标准形式与Galois群中心性等工具。
AI 中文摘要
设$\Sm_q$和$\Rm_q$分别表示Suzuki曲线和Ree曲线。受Giulietti--Korchmáros曲线的启发，Skabelund构造了这些曲线的循环覆盖$\tSm_q$和$\tRm_q$，并证明了它们分别在$\F_{q^4}$和$\F_{q^6}$上是极大的。在同一篇论文中，他将Suzuki曲线和Ree曲线与某些射线类域覆盖$\Sm_{\rm rcf}$和$\Rm_{\rm rcf}$相关联，并证明了存在塔\\[ \Sm_{\rm rcf}\longrightarrow\tSm_q\longrightarrow\Sm_q, \qquad \Rm_{\rm rcf}\longrightarrow\tRm_q\longrightarrow\Rm_q. \\] 对较小的$q$值进行的计算表明，每个塔中的第一个映射总是同构，一般情形当时尚未解决。我们证明对于所有可容许的$q$，有\\[ \Sm_{\rm rcf}=\tSm_q,\qquad \Rm_{\rm rcf}=\tRm_q \\]，其中在Suzuki情形下比较在$\F_{q^4}$上进行，在Ree情形下比较在$\F_{q^6}$上进行。证明结合了射线类扩张的Kummer标准形式（允许常数扭曲）、其Galois群在提升自同构中的中心性、基曲线的标准对合以及无穷远有理点处的第一个正非间隙。
英文摘要
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.
