Let $S_0$ be a smooth (projective?) and (geometrically) connected scheme over a finite field of characteristic $p$ and let $S$ be its base change to an algebraic closure of the finite field. Let $\pi:A \to S_0$ be an abelian scheme of relative dimension $g$ such that the Newton polygon of $A[p^{\infty}]$ is constant. For all $\ell \not=p$ we can consider the local systems $R^1 \pi_{\ast} \mathbb{Q}_{\ell}$ as a representation $\rho_{\ell}$ of $\pi_1^{\text{ét}}(S)$, and deep results of Deligne tell us that this is a semisimple representation; in fact the Zariski closure of the image of $\rho$ is "independent of $\ell$". Similarly there is a a $p$-adic variant $\mathcal{E}$ of $R^1 \pi_{\ast} \mathbb{Q}_{\ell}$, which is an (overconvergent) isocrystal, it also has the same monodromy group (see https://arxiv.org/abs/1711.06669) as its $\ell$-adic cousins.

Let $\mathbb{X}_b$ be a $p$-divisible group over $\overline{\mathbb{F}}_p$ with the same Newton polygon as $A[p^{\infty}]$ and let $J_b(\mathbb{Q}_p) \subset G(\breve{\mathbb{Q}}_p)$ be the automorphism group of the isocrystal associated to $\mathbb{X}_b$. Then Proposition 4.3.13 of Caraini-Scholze (https://arxiv.org/abs/1511.02418) gives us a pro-étale $J_b(\mathbb{Q}_p$)-torsor which roughly speaking parametrises quasi-isogenies $A[p^{\infty}] \to \mathbb{X}_b$.

Question: Is there any relation between the Zariski closure of the image of $\rho':\pi_1^{\text{ét}}(S) \to J_b(\mathbb{Q}_p)$ and the geometric monodromy groups of $\rho_{\ell}$ and $\mathcal{E}$?

Question: Is the Zariski closure of the image of $\rho':\pi_1^{\text{ét}}(S) \to J_b(\mathbb{Q}_p)$ reductive?

Example: If $S_0=Y_1(N)^{\text{ord}}$ is the mod $p$ fiber of the ordinary locus of the modular curve, then the $\ell$-adic monodromy group is equal to $\operatorname{SL}_2$ and the Zariski closure of the image of $\rho'$ is $\mathbb{G}_m=J_b$.


Let me briefly answer your question. There are two $p$-adic analogues of $R^1\pi_*\mathbb{Q}_\ell/S_0$: a (convergent) $F$-isocrystal $\mathcal E$ and an overconvergent $F$-isocrystal $\mathcal{E}^\dagger$. These two objects define algebraic monodromy groups $G_F(\mathcal E)$ and $G_F(\mathcal E^\dagger)$, where the first algebraic group is naturally a subgroup of the other. The algebraic group $G_F(\mathcal E^\dagger)$ (which for simplicity I assume connected) is “the same” as the arithmetic monodromy group of $R^1\pi_*\mathbb{Q}_\ell$ (as proven in the article you quoted). Moreover, one can attach to $G_F(\mathcal E^\dagger)$ a cocharacter $\lambda$ induced by the slope filtration of $\mathcal E$. What I have also proven is that $G_F(\mathcal E)$ is the parabolic subgroup $P(\lambda)\subseteq G_F(\mathcal E^\dagger)$ attached to $\lambda$. As a consequence, if $G(\rho')$ is the Zariski closure of your $\rho'$ and $Z(\lambda)$ is the centraliser of $\lambda$, there is an exact sequence $$1\to G(\rho')\to Z(\lambda)\to T\to 1$$ where $T$ is a torus. In particular, $G(\rho')$ is a reductive group. I gave two talks here in Bonn about a generalisation of this result. You can find some extended notes at http://guests.mpim-bonn.mpg.de/daddezio/MSPC-BN19.pdf. If you have any further questions you can send me an email.


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