Rigidification of arithmetic $\mathcal{D}$-modules and an overconvergent Riemann-Hilbert correspondence

Abstract

In this article, I define triangulated categories of constructible isocrystals on varieties over a perfect field of positive characteristic, in which Le Stum's abelian category of constructible isocrystals sits as the heart of a natural t-structure. I then prove a Riemann-Hilbert correspondence, showing that, for objects admitting some (unspecified) Frobenius, this triangulated category is equivalent to the triangulated category of overholonomic ${\mathcal{D}}^{†}$-modules in the sense of Caro. I also show that the cohomological functors $f^{!}$, $f_{+}$ and $\otimes$ defined for ${\mathcal{D}}^{†}$-modules have natural interpretations on the constructible side of this correspondence. Finally, I use this to prove that, for any variety $X$ admitting an immersion into a smooth and proper formal scheme, rigid cohomology (with lisse coefficients) agrees with cohomology defined using arithmetic $\mathcal{D}$-modules.

84 pages, comments very welcome!