On a torsion analogue of the weight-monodromy conjecture

We formulate and study a torsion analogue of the weight-monodromy conjecture for a proper smooth scheme over a non-archimedean local field. We prove it for proper smooth schemes over equal characteristic non-archimedean local fields, abelian varieties, surfaces, varieties uniformized by Drinfeld upper half spaces, and set-theoretic complete intersections in projective smooth toric varieties. In the equal characteristic case, our methods rely on an ultraproduct variant of Weil II established by Cadoret.