A stacky nilpotent $p$-adic Riemann-Hilbert correspondence

TL;DR

This paper establishes a nilpotent $p$-adic Riemann-Hilbert correspondence for smooth rigid varieties over $C= ext{C}_p$, using stack language to relate local systems and $t$-connections via an equivalence of moduli stacks.

Contribution

It introduces the moduli stacks of $ ext{B}_{dR}^+$-local systems and $t$-connections and proves their equivalence in the nilpotent case, advancing the $p$-adic Riemann-Hilbert theory.

Findings

Established an equivalence of stacks for nilpotent local systems and $t$-connections.

Defined the moduli stacks of $ ext{B}_{dR}^+$-local systems and $t$-connections.

Proved the nilpotent $p$-adic Riemann-Hilbert correspondence using stack language.

Abstract

Let $\overline X$ be a smooth rigid variety over $C=\mathbb C_p$ admitting a lift $X$ over $B_{dR}^+$. In this paper, we use the stacky language to prove a nilpotent $p$-adic Riemann-Hilbert correspondence. After introducing the moduli stack of $\mathbb B^+_{dR}$-local systems and $t$-connections, we prove that there is an equivalence of the nilpotent locus of the two stacks: $RH^0:LS^0_X \to tMIC^0_X$, where $LS^0_X$ is the stack of nilpotent $\mathbb B^+_{dR}$-local systems on $\overline X_{1,v}$ and $tMIC^0_X$ is the stack of $\mathcal{O}_X$-bundles with integrable $t$-connection on $X_{et}$.