Weights in arithmetic geometry

TL;DR

This paper reviews the concept of weights in arithmetic geometry, exploring their origins, various manifestations across cohomology theories, and recent applications to Hasse principles and motivic cohomology.

Contribution

It provides a comprehensive overview of weights in arithmetic geometry, highlighting recent applications and open questions in the field.

Findings

Weights connect different cohomology theories.

Applications include Hasse principles and motivic cohomology computations.

Open questions remain in understanding weights in various contexts.

Abstract

The concept of weights on the cohomology of algebraic varieties was initiated by fundamental ideas and work of A. Grothendieck and P. Deligne. It is deeply connected with the concept of motives and appeared first on the singular cohomology as the weights of (possibly mixed) Hodge structures and on the etale cohomology as the weights of eigenvalues of Frobenius. But weights also appear on algebraic fundamental groups and in p-adic Hodge theory, where they become only visible after applying the comparison functors of Fontaine. After rehearsing various versions of weights, we explain some more recent applications of weights, e.g., to Hasse principles and the computation of motivic cohomology, and discuss some open questions.