# On the semisimplicity of reductions and adelic openness for $E$-rational compatible systems over global function fields

## TL;DR

This paper investigates the properties of geometric monodromy groups of compatible systems over global function fields, establishing semisimplicity and adelic openness results using automorphic methods, with implications for understanding Galois representations.

## Contribution

It proves semisimplicity of the geometric monodromy groups' closures and provides criteria for adelic openness in compatible systems over function fields.

## Key findings

- Semisimplicity of geometric monodromy closures for almost all primes.
- The special fiber matches the Nori envelope of mod-$oldsymbol{ extlambda}$ reductions.
- Criteria for adelic openness of the monodromy image.

## Abstract

Let $X$ be a normal geometrically connected variety over a finite field $\kappa$ of characteristic~$p$. Let $E$ be a number field. Using automorphic methods over global function fields, we derive properties of the geometric monodromy groups of arbitrary connected $E$-rational semisimple compatible systems $(\rho_\lambda)$ of $n$-dimensional representations of the arithmetic fundamental group $\pi_1(X)$, where $\lambda$ ranges over the finite places of $E$ not above $p$: Let $\Lambda_\lambda$ be any $\pi_1(X)$-stable lattice in $E_\lambda^n$ under $\rho_\lambda$. Then for almost all $\lambda$, the schematic closure of the geometric monodromy $\rho_\lambda(\pi_1(X_{\overline{\kappa}}))$ in $\mathrm{Aut}_{\mathcal{O}_\lambda}(\Lambda_\lambda)$ is a semisimple $\mathcal{O}_\lambda$-group scheme, and its special fiber agrees with the Nori envelope of the geometric monodromy of the mod-$\lambda$ reduction of $\rho_\lambda$. A comparable result under different hypotheses was recently proved by Cadoret, Hui and Tamagawa by other methods. We also provide natural criteria for the image of $\pi_1(X_{\overline{\kappa}})$ under $\prod_\lambda\rho_\lambda$ to have adelic open image in an appropriate sense.

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Source: https://tomesphere.com/paper/1901.03654