Images directes et fonctions L en cohomologie rigide

TL;DR

This paper proves partial cases of Berthelot's conjecture on overconvergence of higher direct images in rigid cohomology, with applications to $L$-functions and $p$-adic properties, especially for liftable morphisms and relative complete intersections.

Contribution

It establishes overconvergence of higher direct images under certain conditions, advancing the understanding of rigid cohomology and related $L$-functions in characteristic $p$.

Findings

Overconvergence proven for smooth, liftable, or complete intersection cases.

Rationality and meromorphy of $L$-functions derived from these cohomological results.

Analysis of $p$-adic zeros and poles for specific cases like ordinary abelian schemes.

Abstract

Let $k$ be a perfect field of characteristic $p>0$, $\mathcal{V}$ a complete discrete valuation ring with residue field $k$ and field of fractions $K$ of characteristic 0, and $S$ a separated $k$-scheme of finite type. When $S$ is smooth over $k$, we partially prove here a conjecture of Berthelot about the overconvergence of the higher direct images of the structure sheaf under a proper smooth morphism $f:X\to S$; when $k$ is perfect and $\mathcal{V}$ is tamely ramified such direct images are always convergent, not only for the structure sheaf but also for (almost) every convergent $F$-isocrystals. More generally, we prove this overconvergence when $f$ is liftable over $\mathcal{V}$, or when $X$ is a relative complete intersection in some projective spaces over $S$, and taking as coefficients any overconvergent isocrystals. We then apply these results to $L$-functions with coefficients such direct images with Frobenius structure: we derive rationality or meromorphy for these $L$-functions (Dwork's conjecture), and we study their $p$-adic unit zeroes and poles (Katz's conjecture) ; and explicit case concerns the ordinary abelian schemes. A more precise presentation of results by chapters is given in the introduction.