# A question about $p$-adic monodromy of abelian varieties

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.