Lifting of abelian schemes, \(F\)-isocrystals and \(L\)-functions (Q2720956)

The paper under review uses \(p\)-adic techniques to study certain \(L\)-functions for sheaves on a scheme \(S\) over a finite field. The sheaves are the overconvergent versions of the cohomologies (in different degrees) of a relative abelian scheme over \(S\), with a separable polarisation. The latter hypothesis is used to lift the abelian scheme to characteristic zero which is necessary to construct overconvergent crystals. The method provides an alternative to the usual \(l\)-adic method (invented by Grothendieck) used to deal with such \(L\)-functions.