\(L\)-functions associated to overconvergent \(F\)-isocrystals (Q1318074)

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 \(q^ n\).