A motivic approach to rational $p$-adic cohomologies

Abstract

We survey over some recent applications of motivic homotopy theory in the definition and the study of $p$-adic cohomology theories. In particular, we revisit the proof of the $p$-adic weight-monodromy conjecture for smooth projective hypersurfaces in light of the motivic definition of nearby cycles and monodromy operators.

28 pages, prepared for the "Regulators V" conference, Pisa, June 3-13, 2024