Motivic p-adic periods of 1-motives

TL;DR

This paper constructs an explicit p-adic de Rham comparison isomorphism for 1-motives, introduces a new ring of motivic p-adic periods, and relates these to classical p-adic periods, with applications to explicit computations.

Contribution

It provides a functorial construction of the p-adic de Rham comparison for 1-motives and introduces a novel ring of motivic p-adic periods.

Findings

Constructed an explicit p-adic de Rham comparison isomorphism.

Introduced a new ring of motivic p-adic periods.

Computed period pairings for specific 1-motives and curves.

Abstract

We give an explicit construction of the p-adic de Rham comparison isomorphism for 1-motives. In particular, we prove that our construction recovers the classical de Rham comparison isomorphism and is functorial with respect to morphisms of rigid 1-motives. In the second part, we construct a new ring of motivic p-adic periods using the formalism of rigid analytic motives. Furthermore, we explain the relation between these motivic periods and the periods of the classical p-adic de Rham comparison isomorphism. Finally, we apply our construction to explicitly compute the period pairing for several examples of 1-motives and curves.