The authors construct p-adic integration pairings and Q-structures for 1-motives with good reduction, and claim that all linear relations among the resulting p-adic periods at depths 1 and 2 are captured by bilinearity and functoriality.
Une introduction aux motifs (motifs purs, motifs mixtes, p\'eriodes) , volume 17 of Panoramas et Synth\`eses
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P-adic Period Conjectures for 1-motives: Integration and Linear Relations
The authors construct p-adic integration pairings and Q-structures for 1-motives with good reduction, and claim that all linear relations among the resulting p-adic periods at depths 1 and 2 are captured by bilinearity and functoriality.