Finite subgroups of outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields
Abstract
In the present paper, we study finite subgroups of the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields from the point of view of mono-anabelian geometry. We shall refer to a finite subgroup of the outer automorphism group of the absolute Galois group of a mixed-characteristic local field as quasi-geometric if its inverse image in the automorphism group of the corresponding absolute Galois group is isomorphic to the absolute Galois group of some mixed-characteristic local field. It is well-known that the image of the natural injective homomorphism from the automorphism group of a mixed-characteristic local field to the outer automorphism group of the associated absolute Galois group yields a typical example of a quasi-geometric subgroup. The main result of the present paper asserts the existence of non-quasi-geometric finite subgroups of the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields. This result shows that the set of quasi-geometric subgroups of the outer automorphism group of the absolute Galois group of a mixed-characteristic local field cannot be characterized, in general, as the set of all finite subgroups of the outer automorphism group.
Related
- Large cyclic automorphism groups and cyclic subgroups of index two in characteristic two
- Finite Symplectic Automorphism Groups of K3 Surfaces in Characteristic Seven
- K3 surfaces with finite automorphism group in positive characteristic
- A Quadratic Bound for Automorphism Groups of Curves of Positive p-Rank in Odd Characteristic
- Sharp bounds and gap phenomena for large p-subgroups of automorphism groups of curves
BibTeX
@misc{nishio2026finite,
title = {{Finite subgroups of outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields}},
author = {Yu Nishio},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.19696},
eprint = {2609.19696},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.19696},
}
RIS
TY - UNPB
TI - Finite subgroups of outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields
AU - Yu Nishio
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.19696
UR - https://arxiv.org/abs/2609.19696
AB - In the present paper, we study finite subgroups of the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields from the point of view of mono-anabelian geometry. We shall refer to a finite subgroup of the outer automorphism group of the absolute Galois group of a mixed-characteristic local field as quasi-geometric if its inverse image in the automorphism group of the corresponding absolute Galois group is isomorphic to the absolute Galois group of some mixed-characteristic local field. It is well-known that the image of the natural injective homomorphism from the automorphism group of a mixed-characteristic local field to the outer automorphism group of the associated absolute Galois group yields a typical example of a quasi-geometric subgroup. The main result of the present paper asserts the existence of non-quasi-geometric finite subgroups of the outer automorphism groups of the absolute Galois groups of mixed-characteristic local fields. This result shows that the set of quasi-geometric subgroups of the outer automorphism group of the absolute Galois group of a mixed-characteristic local field cannot be characterized, in general, as the set of all finite subgroups of the outer automorphism group.
ER -