A motivic approach to rational $p$-adic cohomologies

TL;DR

This paper explores how motivic homotopy theory can be applied to define and analyze $p$-adic cohomology theories, offering new insights into the weight-monodromy conjecture for certain algebraic varieties.

Contribution

It introduces a motivic perspective to $p$-adic cohomologies and revisits the proof of the weight-monodromy conjecture using motivic nearby cycles and monodromy operators.

Findings

Motivic approach provides a new framework for $p$-adic cohomology theories.

Revised proof of the $p$-adic weight-monodromy conjecture for smooth projective hypersurfaces.

Enhanced understanding of the role of motivic nearby cycles in $p$-adic contexts.

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.