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Lifting of abelian schemes, \(F\)-isocrystals and \(L\)-functions. (Relèvement de schémas abéliens, \(F\)-isocristaux et fonctions \(L\).) (English) Zbl 0987.14013
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.

MSC:
14F30 \(p\)-adic cohomology, crystalline cohomology
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
14G20 Local ground fields in algebraic geometry
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