We establish a tilting equivalence for rational, homotopy-invariant cohomology theories defined over non-archimedean analytic varieties. More precisely, we prove an equivalence between the categories of motives of rigid analytic varieties over a perfectoid field K of mixed characteristic and over the associated (tilted) perfectoid field K ♭ of equal characteristic. This can be considered as a motivic generalization of a theorem of Fontaine and Wintenberger, claiming that the Galois groups of K and K ♭ are isomorphic.

ABSTRACT. We prove the equivalence between the category RigDM ef et (K, Q) of effective motives of rigid analytic varieties over a perfect complete non-archimedean field K and the category RigDA eff Frobét (K, Q) which is obtained by localizing the category of motives without transfers RigDA ef et (K, Q) over purely inseparable maps. In particular, we obtain an equivalence between RigDM ef et (K, Q) and RigDA ef et (K, Q) in the characteristic 0 case and an equivalence between DM ef et (K, Q) and DA eff Frobét (K, Q) of motives of algebraic varieties over a perfect field K. We also show a relative and a stable version of the main statement.

We construct the dagger realization functor for analytic motives over nonarchimedean fields of mixed characteristic, as well as the Monsky-Washnitzer realization functor for algebraic motives over a discrete field of positive characteristic. In particular, the motivic language on the classicétale site provides a new direct definition of the overconvergent de Rham cohomology and rigid cohomology and shows that their finite dimensionality follows formally from one of Betti cohomology for smooth projective complex varieties.

Inspired by the work of Cherry, we introduce and study a new notion of Brody hyperbolicity for rigid analytic varieties over a non-archimedean field K of characteristic zero. We use this notion of hyperbolicity to show the following algebraic statement: if a projective variety admits a non-constant morphism from an abelian variety, then so does any specialization of it. As an application of this result, we show that the moduli space of abelian varieties is K-analytically Brody hyperbolic in equal characteristic 0. These two results are predicted by the Green–Griffiths–Lang conjecture on hyperbolic varieties and its natural analogues for non-archimedean hyperbolicity. Finally, we use Scholze’s uniformization theorem to prove that the aforementioned moduli space satisfies a non-archimedean analogue of the “Theorem of the Fixed Part” in mixed characteristic.

We define a de Rham cohomology theory for analytic varieties over a valued field K ♭ of equal characteristic p with coefficients in a chosen untilt of the perfection of K ♭ by means of the motivic version of Scholze's tilting equivalence. We show that this definition generalizes the usual rigid cohomology in case the variety has good reduction. We also prove a conjecture of Ayoub yielding an equivalence between rigid analytic motives with good reduction and unipotent algebraic motives over the residue field, also in mixed characteristic.