Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2407.20907 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
Table of Contents:
- Let $\{ρ_{\ell}:\mathrm{Gal}_K\to\mathrm{GL}_n(\mathbb{Q}_{\ell})\}_{\ell}$ be a semisimple compatible system of $\ell$-adic representations of a number field $K$ that is arising from geometry. Let $\textbf{G}_{\ell}\subset\mathrm{GL}_{n,\mathbb{Q}_{\ell}}$ and $\widehat{\underline{G_{\ell}}}\subset\mathrm{GL}_{n,\mathbb{F}_\ell}$ be respectively the algebraic monodromy group and full algebraic envelope of $ρ_{\ell}$. We prove that there is a natural isomorphism between the component groups $π_0(\textbf{G}_{\ell}) \simeq π_0(\widehat{\underline{G_\ell}})$ for all sufficiently large $\ell$.