Linear Relations of p-adic Periods of 1-motives

Abstract

This work aims to develop p-adic analogs of known results for classical periods, focusing specifically on 1-motives. We establish an integration theory for 1-motives with good reductions, which generalizes the Colmez-Fontaine-Messing p-adic integration for abelian varieties with good reductions. We also compare the integration pairing with other pairings such as those induced by crystalline theory. Additionally, we introduce a formalism for periods and formulate p-adic period conjectures related to p-adic periods arising from this integration pairing. Broadly, our p-adic period conjecture operates at different depths, with each depth revealing distinct relations among the p-adic periods. Notably, the classical period conjecture (Kontsevich-Zagier conjecture over \bar{Q}) for 1-periods fits within our framework, and, according to the classical subgroup theorem of Huber-Wüstholz for 1-motives, the conjecture for classical periods of 1-motives holds true at depth 1. Finally, we identify three Q-structures arising from \bar{Q}-rational points of the formal p-divisible group associated with a 1-motive M with a good reduction at p, and we prove p-adic period conjectures at depths 2 and 1, relative to periods induced by the p-adic integration of M and these Q-structures. Our proof involves a p-adic version of the subgroup theorem that we obtain for 1-motives with good reductions.