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

Abstract:

Let $X$ be a normal geometrically connected variety over a finite field $\kappa$ of characteristic $p$. Let $(\rho _\lambda \colon \pi _1(X)\to \textrm {GL}_n(E_\lambda ) )_\lambda$ be any semisimple $E$-rational compatible system, where $E$ is a number field and $\lambda$ ranges over the finite places of $E$ not above $p$. We derive new properties on the monodromy groups of such systems for almost all $\lambda$ and give natural criteria for the corresponding geometric adelic representation to have open image in an appropriate sense. Key inputs to our results are automorphic methods and the Langlands correspondence over global function fields proved by L. Lafforgue.

To say more, let $(\overline {\rho }_\lambda \colon \pi _1(X)\to \textrm {GL}_n(k_\lambda ) )_\lambda$ be the corresponding mod-$\lambda$ system, where for every $\lambda$ by ${\mathcal O}_\lambda$ and $k_\lambda$ we denote the valuation ring and the residue field of $E_\lambda$, and where the reduction is done with respect to some $\pi _1(X)$-stable ${\mathcal O}_\lambda$-lattice $\Lambda _\lambda$ of $E_\lambda ^n$. Let also $G_\lambda ^{\mathrm {geo}}$ be the Zariski closure of $\rho _\lambda (\pi _1(X_{\overline {\kappa }}))$ in $\textrm {GL}_{n, E}$, and let $\mathcal {G}_\lambda ^{\mathrm {geo}}$ be its schematic closure in $\textrm {Aut}_{{\mathcal O}_\lambda }(\Lambda _\lambda )$. Assume in the following that the algebraic groups $G_\lambda ^{\mathrm {geo}}$ are connected.

We prove that for almost all $\lambda$ the group scheme $\mathcal {G}_\lambda ^{\mathrm {geo}}$ is semisimple over ${\mathcal O}_\lambda$, and its special fiber agrees with the Nori envelope of $\overline {\rho }_\lambda (\pi _1(X_{\overline {\kappa }}))$. A comparable result under different hypotheses was proved by A. Cadoret, C.-Y. Hui, and A. Tamagawa using other methods. As an intermediate result, we show for $X$ a curve that any potentially tame compatible system of mod-$\lambda$ representations can be lifted to a compatible system over a number field; this implies for almost all $\lambda$ the semisimplicity of the restriction $\overline {\rho }_\lambda |_{\pi _1(X_{\overline {\kappa }})}$. Finally, we establish adelic openness for $(\rho _\lambda |_{\pi _1(X_{\overline \kappa })} )_\lambda$ in the sense of C. Y. Hui and M. Larsen, for $E={\mathbb {Q}}$ in general, and for $E\supsetneq {\mathbb {Q}}$ under additional hypotheses.