Slopes and weights of $\ell$-adic cohomology of rigid spaces

TL;DR

This paper establishes algebraic properties and bounds for Frobenius eigenvalues in $\, ext{l}$-adic cohomology of rigid spaces over $p$-adic fields, confirming conjectures and providing new examples of monodromy behavior.

Contribution

It proves algebraicity and valuation bounds for Frobenius eigenvalues, confirming conjectures and constructing examples of monodromy-pure sheaves with non-pure cohomology.

Findings

Frobenius eigenvalues are algebraic integers.

Bounds are established for their $p$-adic valuations.

Examples of non-monodromy-pure cohomology are provided.

Abstract

We prove that Frobenius eigenvalues of $\ell$-adic cohomology and $\ell$-adic intersection cohomology of rigid spaces over $p$-adic local fields are algebraic integers and we give bounds for their $p$-adic valuations. As an application, we deduce bounds for their weights, proving conjectures of Bhatt, Hansen, and Zavyalov. We also give examples of monodromy-pure perverse sheaves on projective curves with non monodromy-pure cohomology, answering a question of Hansen and Zavyalov.