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We prove a number of results on the \'etale cohomology of rigid analyticvarieties over $p$-adic non-archimedean local fields. Among other things, weestablish bounds for Frobenius eigenvalues, show a strong version ofGrothendieck's local monodromy theorem, prove mixedness of the nearby cyclesheaf, and show that for any formal model, the IC sheaf on the special fiber iscaptured by the nearby cycles of the IC sheaf on the generic fiber. We alsoprove a local version of Deligne's weight-monodromy conjecture, by a novelperfectoid analysis of nearby cycles.  Along the way, we develop the theory of "constructible $\ell$-adic complexeson Deligne's topos" (six operations, perverse t-structure, a notion ofmixedness, etc.), which is prerequisite to a precise discussion of the Galoisaction on nearby cycles for algebraic and rigid analytic varieties overnon-archimedean fields.

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