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Relative Tate-Shafarevich kernel of a reductive group over a global field
Abstract
We describe an algorithm and a computer program computing the relative Tate-Shafarevich kernel ShaS(K,G) of a reductive group G over a global field K with respect to a finite set S of places of K containing no archimedean places. We assume that we know the algebraic fundamental group Λ=π1alg(G), the Galois group Θ=Gal(L/K) acting on Λ, where L is a finite Galois extension of K having no real places, the decomposition subgroups Θv⊆ Θ for all places v in S, and all non-cyclic decomposition subgroups Θv for v outside S. We also compute ShaS(K,G) in some cases where S contains archimedean places.
Related
- Relative Geometric Invariant Theory: Reductive and Non-reductive
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- A Calabi--Matsushima decomposition for real reductive group actions
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BibTeX
@misc{borovoi2026relative,
title = {{Relative Tate-Shafarevich kernel of a reductive group over a global field}},
author = {Mikhail Borovoi},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.04773},
eprint = {2611.04773},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.04773},
}
RIS
TY - UNPB
TI - Relative Tate-Shafarevich kernel of a reductive group over a global field
AU - Mikhail Borovoi
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.04773
UR - https://arxiv.org/abs/2611.04773
AB - We describe an algorithm and a computer program computing the relative Tate-Shafarevich kernel Sha^S(K,G) of a reductive group G over a global field K with respect to a finite set S of places of K containing no archimedean places. We assume that we know the algebraic fundamental group Λ=π_1^alg(G), the Galois group Θ=Gal(L/K) acting on Λ, where L is a finite Galois extension of K having no real places, the decomposition subgroups Θ_v⊆ Θ for all places v in S, and all non-cyclic decomposition subgroups Θ_v for v outside S. We also compute Sha^S(K,G) in some cases where S contains archimedean places.
ER -