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\(L\)-functions associated to overconvergent \(F\)-isocrystals. (Fonctions \(L\) associées aux \(F\)-isocristaux surconvergents.) (French) Zbl 0789.14015

We prove the formulas that give the cohomological expressions of the \(L\)- function attached to an overconvergent \(F\)-isocristal. We deduce that such an \(L\)-function is always meromorphic. This applies in particular to the \(L\)-functions that arise in the study of classical exponential sums. We also show that Frobenius is always bijective on rigid cohomology with or without support and that the trace of Frobenius on the highest rigid cohomology space with compact support of a scheme of pure dimension \(n\) is multiplication by \(q^ n\).

MSC:

14F30 \(p\)-adic cohomology, crystalline cohomology
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
14F20 Étale and other Grothendieck topologies and (co)homologies
11L03 Trigonometric and exponential sums (general theory)
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References:

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