1-Motives : origins and (p-adic) realizations

Following we explain how 1-motives originate via the simplicial Picard functorof varieties defined over a perfect field with special interest for positive characteris-tics. We then consider the crystalline realization of Deligne’s 1-motives in positivecharacteristics and prove a comparison theorem with the De Rham realization of(formal) liftings to zero characteristic. This is proven using the crystalline na-ture of the universal extension of a 1-motive. Using we prove that this resultholds also for general bases. This allows to define a canonical Gauss-Manin con-nection on the de Rham realization of 1-motives. As an application we provide aconstruction of the so called Manin’s map from a motivic point of view.