Etale descent for overconvergent F-isocrystals and rationality of L-functions of Abelian schemes

Let k be a field of characteristic p > 0 and S a smooth separated k-scheme; three types of results are proved: (1) etale descent for overconvergent F-isocrystals on S; (2) full faithfulness of restriction functors to a dense open subscheme of S, between F-isocrystals; (3) a p-adic cohomological expression of the L-function of an abelian scheme on S and its rationality. (C) 2002 Editions scientitiques et medicales Elsevier SAS.