Height functions for motives

TL;DR

This paper introduces new height functions for motives over number fields, compares them with classical and Hodge-theoretic heights, and explores implications and open questions in the field.

Contribution

It defines and compares various height functions for motives, linking them to classical and Hodge-theoretic heights, and raises new questions in the study of motives.

Findings

New height functions for motives are introduced.

Comparisons with classical and Hodge-theoretic heights are established.

The work suggests new research directions and questions in motive theory.

Abstract

We define various height functions for motives over number fields. We compare these height functions with classical height functions on algebraic varieties, and also with analogous height functions for variations of Hodge structures on curves over C. These comparisons provide new questions on motives over number fields.