Linear relations of p-adic periods of 1-motives (thesis)

TL;DR

This thesis develops p-adic analogs of classical period results for 1-motives, establishing an integration theory, formulating p-adic period conjectures, and proving these conjectures at certain depths using a p-adic subgroup theorem.

Contribution

It introduces a p-adic integration framework for 1-motives, formulates p-adic period conjectures, and proves them at specific depths, extending classical period theory into the p-adic setting.

Findings

Established a p-adic integration theory for 1-motives with good reduction.

Formulated p-adic period conjectures related to this integration pairing.

Proved p-adic period conjectures at depths 1 and 2 for 1-motives.

Abstract

In this thesis, we aim 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-Zanier conjecture over $\bar{\mathbb{Q}}$) for 1-periods fits within our framework, and, according to the classical subgroup theorem of Huber-W\"ustholz for 1-motives, the conjecture for classical periods of 1-motives holds true at depth 1. Finally, we identify three $\mathbb{Q}$-structures arising from $\bar{\mathbb{Q}}$-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 $\mathbb{Q}$-structures. Our proof involves a p-adic version of the subgroup theorem that we obtain for 1-motives with good reductions.