Height of motives

TL;DR

This paper introduces a new way to measure the complexity of motives over number fields and links the finiteness of motives of bounded height to progress on the Tate conjecture for motives with good reduction.

Contribution

It defines the height of a motive over a number field and connects the finiteness of motives of bounded height to the Tate conjecture for motives with good reduction.

Findings

Finiteness of motives of bounded height implies Tate conjecture for certain motives.

Provides a new invariant (height) for motives over number fields.

Establishes a conditional link between motive classification and deep conjectures.

Abstract

We define the height of a motive over a number field. We show that if we assume the finiteness of motives of bounded height, Tate conjecture for the $p$-adic Tate module can be proved for motives with good reduction at $p$.