Height pairings of 1-motives

TL;DR

This paper generalizes p-adic height pairings to 1-motives, establishing a global pairing between rational points and their duals, and providing local pairings through Picard and Albanese 1-motives.

Contribution

It introduces a new framework for p-adic height pairings in the context of 1-motives, extending previous work on abelian varieties.

Findings

Defined a global pairing between rational points of 1-motives and their duals.

Constructed local pairings using Picard and Albanese 1-motives.

Extended the concept of height pairings to a broader class of motives.

Abstract

The purpose of this work is to generalize, in the context of 1-motives, the $p$-adic height pairings constructed by B. Mazur and J. Tate on abelian varieties. Following their approach, we define a global pairing between the rational points of a 1-motive and its dual. We also provide local pairings between zero-cycles and divisors on a curve, which is done by considering its Picard and Albanese 1-motives.