Picard 1-motives and motivic cohomoloy of varieties in characteristic p.

I will present joint work with L. Barbieri-Viale (Rome I). Given a variety V in characteristic 0, possibly open and singular, Barbieri-Viale and Srinivas have associated a Picard 1-motive with ℓ-adic, de Rham and Hodge realizations which coincide with the first ℓ-adic, de Rham and Betti cohomology group of V. I will show how to extend their construction to varieties in any characteristic, using J. de Jong’s theory of alterations. I will recall a classical definition of a p-adic realization of a 1-motive in characteristic p. I will prove that if V is a variety in characteristic p, then the p-adic realization of the associated Picard 1-motive coincides with a suitable p-adic cohomology group defined directly on V as proposed by de Jong.