Arithmetic Properties Of $\ell$-adic \'Etale Cohomology and Nearby Cycles of Rigid-Analytic Spaces

TL;DR

This paper advances the understanding of e9tale cohomology in rigid-analytic geometry over p-adic fields, establishing bounds, monodromy, mixedness, and weight conjectures through novel perfectoid techniques.

Contribution

It introduces a new framework for constructible e9tale complexes on Deligne's topos and proves key properties of nearby cycles, including a local weight-monodromy conjecture.

Findings

Bounds for Frobenius eigenvalues established

Strong version of Grothendieck's local monodromy theorem proved

Mixedness of the nearby cycle sheaf demonstrated

Abstract

We prove a number of results on the \'etale cohomology of rigid analytic varieties over $p$-adic non-archimedean local fields. Among other things, we establish bounds for Frobenius eigenvalues, show a strong version of Grothendieck's local monodromy theorem, prove mixedness of the nearby cycle sheaf, and show that for any formal model, the IC sheaf on the special fiber is captured by the nearby cycles of the IC sheaf on the generic fiber. We also prove a local version of Deligne's weight-monodromy conjecture, by a novel perfectoid analysis of nearby cycles. Along the way, we develop the theory of "constructible $\ell$-adic complexes on Deligne's topos" (six operations, perverse t-structure, a notion of mixedness, etc.), which is prerequisite to a precise discussion of the Galois action on nearby cycles for algebraic and rigid analytic varieties over non-archimedean fields.