Hidden parabolic subgroups of GLn over finite chain rings
Abstract
We give a bounded-error quantum polynomial-time algorithm for finding hidden parabolic subgroups of GLn(R) over any effectively presented finite chain ring R. We call auxiliary group elements or subgroups used to extract information through oracle queries \emph{probes}. For chain length at least two, elementary abelian socle probes first recover the complete flag modulo the maximal ideal. When the residue field has size q≥3, we reconstruct the flag recursively, lifting rank-one terminal members one digit at a time. The intermediate digits reduce to abelian hidden subgroup problems, while the final digit is a finite-field multiple-shift problem solved using learning from subset and Fourier sampling. For q=2, where distinct flags can have the same stabilizer, we instead lift the conjugate parabolic subgroup through the congruence filtration. A final linear system determines it modulo normalizer directions. The algorithm outputs generators with probability at least 1−ε in time poly(n,log|R|,log(1/ε)). Consequences include GLn(ℤ/Nℤ) for every N≥2 and, with an explicit maximal-order presentation, GLn(𝒪K/N𝒪K) for number fields K.
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BibTeX
@misc{imran2026hidden,
title = {{Hidden parabolic subgroups of $\mathrmGL_n$ over finite chain rings}},
author = {Muhammad Imran},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28368},
eprint = {2610.28368},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28368},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Hidden parabolic subgroups of GL_n over finite chain rings
AU - Muhammad Imran
PY - 2026
DA - 2026/10/21
PB - arXiv
DO - 10.48550/arxiv.2610.28368
UR - https://arxiv.org/abs/2610.28368
AB - We give a bounded-error quantum polynomial-time algorithm for finding hidden parabolic subgroups of GL_n(R) over any effectively presented finite chain ring R. We call auxiliary group elements or subgroups used to extract information through oracle queries \emph{probes}. For chain length at least two, elementary abelian socle probes first recover the complete flag modulo the maximal ideal. When the residue field has size q≥3, we reconstruct the flag recursively, lifting rank-one terminal members one digit at a time. The intermediate digits reduce to abelian hidden subgroup problems, while the final digit is a finite-field multiple-shift problem solved using learning from subset and Fourier sampling. For q=2, where distinct flags can have the same stabilizer, we instead lift the conjugate parabolic subgroup through the congruence filtration. A final linear system determines it modulo normalizer directions. The algorithm outputs generators with probability at least 1−ε in time poly(n,log|R|,log(1/ε)). Consequences include GL_n(ℤ/Nℤ) for every N≥2 and, with an explicit maximal-order presentation, GL_n(𝒪_K/N𝒪_K) for number fields K.
ER -