On Skabelund's Ray Class Field Covers of the Suzuki and Ree Curves
Abstract
Let Smq and Rmq denote the Suzuki and Ree curves. Motivated by the Giulietti--Korchmáros curve, Skabelund constructed cyclic covers tSmq and tRmq of these curves and proved that they are maximal over Fq4 and Fq6, respectively. In the same paper he associated to the Suzuki and Ree curves certain ray class field covers Smrcf and Rmrcf, and showed that there are towers Smrcf⟶tSmq⟶Smq, Rmrcf⟶tRmq⟶Rmq . 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 Smrcf=tSmq, Rmrcf=tRmq for every admissible q, the comparison being made over Fq4 in the Suzuki case and over Fq6 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.
Related
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BibTeX
@misc{tafazolian2026skabelunds,
title = {{On Skabelund's Ray Class Field Covers of the Suzuki and Ree Curves}},
author = {Saeed Tafazolian},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.13092},
eprint = {2609.13092},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.13092},
}
RIS
TY - UNPB
TI - On Skabelund's Ray Class Field Covers of the Suzuki and Ree Curves
AU - Saeed Tafazolian
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.13092
UR - https://arxiv.org/abs/2609.13092
AB - 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_rcf and Rm_rcf, and showed that there are towers Sm_rcf⟶tSm_q⟶Sm_q, Rm_rcf⟶tRm_q⟶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_rcf=tSm_q, 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.
ER -