Hasse principle for Rost motives

TL;DR

This paper establishes a Hasse principle for certain motives associated with smooth projective quadrics over number fields, showing they are related to Rost motives and providing a complete motivic decomposition in specific cases.

Contribution

It proves a Hasse principle for binary summands of Chow motives of quadrics and characterizes these summands as twists of Rost motives, with a full decomposition when F has at most one real embedding.

Findings

Hasse principle holds for binary summands of Chow motives of quadrics

Such summands are twists of Rost motives

Complete motivic decomposition described for fields with at most one real embedding

Abstract

We prove a Hasse principle for binary direct summands of the Chow motive of a smooth projective quadric Q over a number field F. Besides, we show that such summands are twists of Rost motives. In the case when F has at most one real embedding we describe a complete motivic decomposition of Q.