P-adic Period Conjectures for 1-motives: Integration and Linear Relations

We develop a $p$-adic theory of periods for 1-motives, extending the classical theory of complex periods into the non-archimedean setting. For 1-motives with good reduction over $p$-adic local fields, we construct a $p$-adic integration pairing that generalizes the Colmez--Fontaine--Messing theory for abelian varieties. This pairing is bilinear, perfect, Galois-equivariant, and compatible with the Hodge filtration, taking values in a quotient of the de Rham period ring. Building on this construction, we introduce a stratified formalism for $p$-adic periods, defining period spaces at various depths that capture increasingly refined relations among periods, and formulating conjectures that mirror the Grothendieck period conjecture in this new context. The classical period conjecture for 1-motives over $\bar{Q}$, previously resolved via the Huber--Wustholz analytic subgroup theorem, is recovered at depth 1 in our framework. For 1-motives over number fields with good reduction at $p$, we identify canonical $Q$-structures on the $p$-adic realizations arising from rational points of their associated formal $p$-divisible groups. Relative to these structures, we establish the conjectures at depths 1 and 2. A key tool is the development of a $p$-adic analytic subgroup theorem tailored to 1-motives, providing an analogue of Wustholz's classical result. Our work not only yields a $p$-adic counterpart to the Kontsevich--Zagier conjecture for 1-motives but also opens new pathways for the study of linear relations among $p$-adic periods and their transcendence properties.