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Automorphic Galois representations and the inverse Galois problem for certain groups of type $ D_{m}$


Author: Adrián Zenteno
Journal: Proc. Amer. Math. Soc. 149 (2021), 89-95
MSC (2010): Primary 11F80; Secondary 12F12, 20G40
DOI: https://doi.org/10.1090/proc/15253
Published electronically: October 20, 2020
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Abstract: Let $ m$ be an integer greater than three and $ \ell $ be an odd prime. In this paper we prove that at least one of the following groups: $ \mathrm {P}\Omega ^\pm _{2m}(\mathbb{F}_{\ell ^s})$, $ \mathrm {PSO}^\pm _{2m}(\mathbb{F}_{\ell ^s})$, $ \mathrm {PO}_{2m}^\pm (\mathbb{F}_{\ell ^s})$, or $ \mathrm {PGO}^\pm _{2m}(\mathbb{F}_{\ell ^s})$ is a Galois group of $ \mathbb{Q}$ for infinitely many integers $ s > 0$. This is achieved by making use of a slight modification of a group theory result of Khare, Larsen, and Savin, and previous results of the author on the images of the Galois representations attached to cuspidal automorphic representations of $ \mathrm {GL}_{2m}(\mathbb{A}_\mathbb{Q})$.


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Additional Information

Adrián Zenteno
Affiliation: Instituto de Matemáticas, Pontificia Universidad Católica de Valparaíso, Blanco Viel 596, Cerro Barón, Valparaíso, Chile
Email: adrian.zenteno@pucv.cl

DOI: https://doi.org/10.1090/proc/15253
Received by editor(s): November 15, 2019
Received by editor(s) in revised form: May 18, 2020, and May 31, 2020
Published electronically: October 20, 2020
Additional Notes: The author was supported by CONICYT Proyecto FONDECYT Postdoctorado No. 3190474.
Communicated by: Romyar Sharifi
Article copyright: © Copyright 2020 American Mathematical Society