A -adic monodromy theorem for de Rham local systems

Abstract

We study horizontal semistable and horizontal de Rham representations of the absolute Galois group of a certain smooth affinoid over a $p$ -adic field. In particular, we prove that a horizontal de Rham representation becomes horizontal semistable after a finite extension of the base field. As an application, we show that every de Rham local system on a smooth rigid analytic variety becomes horizontal semistable étale locally around every classical point. We also discuss potentially crystalline loci of de Rham local systems and cohomologically potentially good reduction loci of smooth proper morphisms.

Local constancy of pro‐unipotent Kummer maps

“…x is an isomorphism is [32,Lemma 4.4]. That the square commutes is obvious, and this then implies that 𝜓 pst x is also an isomorphism.…”

“…Then: Proof. For the first point, Shimizu's 𝑝-adic monodromy theorem [32,Theorem 7.4] implies that 𝐸 is potentially horizontal semistable, meaning that the natural map 𝖡 ∇ st (𝑅) ⊗ ℚ nr 𝑝 𝖣 ∇ pst (𝐸) → 𝖡 ∇ st (𝑅) ⊗ ℚ 𝑝 𝐸 is an isomorphism. This directly gives the equality of dimensions, and by tensoring up to 𝖡 ∇ dR (𝑅) and taking 𝐺 𝑅 𝐿 -fixed points, also the isomorphy of the claimed map.…”

“…There is a tautological action of $$G_{R_K}$$ on $$\overline{V}$$ from the right, which is the same as the usual action of the fundamental group $$G_{R_K}$$ on the pointed universal covering $$\overline{V}$$. Following [32, p. 47], we write $$$$\begin{equation*} {\mathsf {B}}_{\rm dR}(R):={\mathcal O}{\mathbb {B}}_{{\rm dR},V}(\overline{V}) \,, \end{equation*}$$$$which is a filtered $$\overline{R}[p^{-1}]$$‐algebra endowed with an action of $$G_{R_K}$$ extending the tautological action on $$\overline{R}[p^{-1}]$$, and with a connection $$$$\begin{equation*} \nabla \colon {\mathsf {B}}_{\rm dR}(R) \rightarrow \Omega ^{1,f}_{R[p^{-1}]/K}\otimes _{R[p^{-1}]}{\mathsf {B}}_{\rm dR}(R) \end{equation*}$$$$satisfying the Leibniz rule with respect to the derivation on …”

“…We write $${\mathsf {B}}_{\rm dR}^\nabla (R)$$ for the kernel of the connection. The $$G_{R_K}$$‐invariant subring of $${\mathsf {B}}_{\rm dR}^\nabla (R)$$ is $$K$$ [32, Proposition 4.9].…”

“…It is more or less clear that Theorem 1.4(2) implies point (a) above; what deserves a little more explanation is how it implies point (b). The extra ingredient required is theory developed by Shimizu [32], which shows that for a horizontal de Rham local system $${\mathbb {E}}$$ on a closed polydisc or spherical polyannulus $$V$$, the isomorphism class of $${\mathsf {D}}_{\rm pst}({\mathbb {E}}_{\bar{y}})$$ as a $$(\varphi ,N,G_K)$$‐module is constant as $$y$$ ranges over $$K$$‐points of $$V$$. Applying this theory to $${}_x{\mathbb {E}}^\mathrm{\acute{e}t}$$ then gives the local constancy required for Theorem 1.2.…”