Constructible nabla-modules on curves

TL;DR

This paper introduces constructible convergent bla-modules on curves over a discrete valuation ring, establishing an equivalence with certain -modules when a Frobenius lift exists, extending the theory of p-adic differential equations.

Contribution

It defines constructible convergent bla-modules on curves and proves an equivalence with perverse holonomic -modules under Frobenius lifting.

Findings

Constructible bla-modules generalize coherent modules on curves.

A specialization functor relates these modules to ^\u00a0modules.

Equivalence holds between constructible F-bla-modules and perverse holonomic F-^\u00a0modules with Frobenius lift.

Abstract

Let $\mathcal V$ be a discrete valuation ring of mixed characteristic with perfect residue field. Let $X$ be a geometrically connected smooth proper curve over $\mathcal V$. We introduce the notion of constructible convergent $\nabla$-module on the analytification $X_{K}^{\mathrm{an}}$ of the generic fibre of $X$. A constructible module is an $\mathcal O_{X_{K}^{\mathrm{an}}}$-module which is not necessarily coherent, but becomes coherent on a stratification by locally closed subsets of the special fiber $X_{k}$ of $X$. The notions of connection, of (over-) convergence and of Frobenius structure carry over to this situation. We describe a specialization functor from the category of constructible convergent $\nabla$-modules to the category of $\mathcal D^\dagger_{\hat X \mathbf Q}$-modules. We show that if $X$ is endowed with a lifting of the absolute Frobenius of $X$, then specialization induces an equivalence between constructible $F$-$\nabla$-modules and perverse holonomic $F$-$\mathcal D^\dagger_{\hat X \mathbf Q}$-modules.