TL;DR
This paper explores the relationship between heights and Tamagawa numbers of motives, providing partial answers to Kato's questions and concrete computations for specific cases of mixed Tate motives.
Contribution
It introduces a connection between heights and Tamagawa numbers of motives and offers explicit calculations for certain classes of mixed Tate motives.
Findings
Partial answer to Kato's question on bounded heights of mixed motives
Concrete computation of the number of mixed Tate motives with three graded quotients
Establishment of a relationship between heights and Tamagawa numbers
Abstract
K. Kato has recently defined and studied heights of mixed motives and proposed some interesting questions. In this paper, we relate the study of heights to the study of Tamagawa numbers of motives. We also partially answer one of Kato's questions about the number of mixed motives of bounded heights in the case of mixed Tate motives with two graded quotients. Finally, we provide a concrete computation with the number of mixed Tate motives with three graded quotients.
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