On the generalized Ree curve
Abstract
Let p be an odd prime, let q_0 := p^s with s >= 1, and put q := p q_0^2. We study the smooth projective curve R_{p,s} with function field F_q(x,y,z), y^q - y = x^{q_0}(x^q - x), z^q - z = x^{2q_0}(x^q - x). The two-equation layer is already present in the ray-class and big-action literature; for p = 3 it gives the classical Ree curve. In analogy with the generalized Suzuki curve, we call R_{p,s} the generalized Ree curve associated with p and s. For p > 3 our principal geometric result determines the full geometric automorphism group: Aut_{F_q}(R_{p,s}) is isomorphic to the semidirect product of U and F_q^*, with |U| = q^3. The natural p-group U gives a big action; we determine the complete ramification filtration of the elementary abelian cover R_{p,s} -> P^1_x and prove that (R_{p,s}, P_infinity) is Castle. In characteristic 3 the same subgroup is only the stabilizer of P_infinity in the full Ree group, so the automorphism structure exhibits a sharp characteristic-three dichotomy. On the arithmetic side, writing Y_{p,s} : y^q - y = x^{q_0}(x^q - x), Z_{p,s} : z^q - z = x^{2q_0}(x^q - x), we prove the F_q-isogeny Jac(R_{p,s}) ~ Jac(Y_{p,s}) x Jac(Z_{p,s})^q. The first factor is supersingular and all three curves have p-rank zero. For p > 3 the first Newton slope of Z_{p,s} satisfies 1/(pq_0 + 2) <= lambda_min(Z_{p,s}) <= 1/3, so Z_{p,s} and R_{p,s} are not supersingular. For fixed p > 3 and s -> infinity, the order of the full automorphism group is asymptotic to 2^(8/5) p^(-4/5) q^(8/5).
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BibTeX
@misc{kazemifard2026generalized,
title = {{On the generalized Ree curve}},
author = {Ahmad Kazemifard and Saeed Tafazolian},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.33912},
eprint = {2609.33912},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.33912},
}
RIS
TY - UNPB
TI - On the generalized Ree curve
AU - Ahmad Kazemifard
AU - Saeed Tafazolian
PY - 2026
DA - 2026/09/27
PB - arXiv
DO - 10.48550/arxiv.2609.33912
UR - https://arxiv.org/abs/2609.33912
AB - Let p be an odd prime, let q_0 := p^s with s >= 1, and put q := p q_0^2. We study the smooth projective curve R_{p,s} with function field F_q(x,y,z), y^q - y = x^{q_0}(x^q - x), z^q - z = x^{2q_0}(x^q - x). The two-equation layer is already present in the ray-class and big-action literature; for p = 3 it gives the classical Ree curve. In analogy with the generalized Suzuki curve, we call R_{p,s} the generalized Ree curve associated with p and s. For p > 3 our principal geometric result determines the full geometric automorphism group: Aut_{F_q}(R_{p,s}) is isomorphic to the semidirect product of U and F_q^*, with |U| = q^3. The natural p-group U gives a big action; we determine the complete ramification filtration of the elementary abelian cover R_{p,s} -> P^1_x and prove that (R_{p,s}, P_infinity) is Castle. In characteristic 3 the same subgroup is only the stabilizer of P_infinity in the full Ree group, so the automorphism structure exhibits a sharp characteristic-three dichotomy. On the arithmetic side, writing Y_{p,s} : y^q - y = x^{q_0}(x^q - x), Z_{p,s} : z^q - z = x^{2q_0}(x^q - x), we prove the F_q-isogeny Jac(R_{p,s}) ~ Jac(Y_{p,s}) x Jac(Z_{p,s})^q. The first factor is supersingular and all three curves have p-rank zero. For p > 3 the first Newton slope of Z_{p,s} satisfies 1/(pq_0 + 2) <= lambda_min(Z_{p,s}) <= 1/3, so Z_{p,s} and R_{p,s} are not supersingular. For fixed p > 3 and s -> infinity, the order of the full automorphism group is asymptotic to 2^(8/5) p^(-4/5) q^(8/5).
ER -