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 $\mathbb{G}_m=J_b$.


1 Answer 1


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.

EDIT: Here is the article about the results of this post: https://arxiv.org/abs/2012.12879 (see Proposition 5.1.4).


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.