Unit Indices of Shanks Orders
Abstract
For an integer t≥−1, let θt be the largest real root of gt(X)=X3−tX2−(t+3)X−1, and set Rt=ℤ[θt]⊆𝒪t=𝒪ℚ(θt), Nt=[𝒪t:Rt], and εt=[𝒪t×:Rt×]. We determine εt when Nt is squarefree, $27$, or $343$: the only nontrivial indices in these cases are ε3=3, ε5=7, ε12=13, and ε54=19. Local conductor calculations give εt| Nt for squarefree Nt. For arbitrary Nt, a regulator comparison shows that t≥2Nt implies εt=1. When Nt=p3 with p≠3 a rational prime, this bound and the index criterion leave at most four possible parameters with εt>1 for each fixed p. For arbitrary additive index, we also prove that 13|εt if and only if t=12 or $66$, with unit index $13$ in both cases. The proof determines the rational solutions of a plane quartic equation by an explicit genus-two descent and a local Chabauty argument, proving Louboutin's Conjecture 19 on its integral solutions. When Nt is a rational prime, we determine the Picard kernel, the cardinality and fibers of the ideal class monoid over Pic(𝒪t), and the corresponding integral matrix-conjugacy classes.
Related
- Proof of a Conjecture of Matherne, Morales, and Selover on Encodings of Unit Interval Orders
- From Unit Interval Orders to Totally Nonnegative Matrices: Permanental Inequalities and Bruhat-Increasing Bijections
- Polynomial Counterexamples to Shanks' Conjecture in Dirichlet-Type Spaces over the Bidisc
- Element Orders and Character Codegrees of Finite Groups
- Higher-order convex orders under free multiplicative convolution
BibTeX
@misc{lu2026unit,
title = {{Unit Indices of Shanks Orders}},
author = {Junyu Lu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2608.30637},
eprint = {2608.30637},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30637},
}
RIS
TY - UNPB
TI - Unit Indices of Shanks Orders
AU - Junyu Lu
PY - 2026
DA - 2026/09/08
PB - arXiv
DO - 10.48550/arxiv.2608.30637
UR - https://arxiv.org/abs/2608.30637
AB - For an integer t≥−1, let θ_t be the largest real root of g_t(X)=X^3−tX^2−(t+3)X−1, and set R_t=ℤ[θ_t]⊆𝒪_t=𝒪_ℚ(θ_t), N_t=[𝒪_t:R_t], and ε_t=[𝒪_t^×:R_t^×]. We determine ε_t when N_t is squarefree, $27$, or $343$: the only nontrivial indices in these cases are ε_3=3, ε_5=7, ε_12=13, and ε_54=19. Local conductor calculations give ε_t| N_t for squarefree N_t. For arbitrary N_t, a regulator comparison shows that t≥2N_t implies ε_t=1. When N_t=p^3 with p≠3 a rational prime, this bound and the index criterion leave at most four possible parameters with ε_t>1 for each fixed p. For arbitrary additive index, we also prove that 13|ε_t if and only if t=12 or $66$, with unit index $13$ in both cases. The proof determines the rational solutions of a plane quartic equation by an explicit genus-two descent and a local Chabauty argument, proving Louboutin's Conjecture 19 on its integral solutions. When N_t is a rational prime, we determine the Picard kernel, the cardinality and fibers of the ideal class monoid over Pic(𝒪_t), and the corresponding integral matrix-conjugacy classes.
ER -