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This paper is motivated by Deninger's programme. First we prove, usingAlvarez Lopez-Kordyukov results, an Atiyah-Bott-Lefschetz trace formula for thecohomology groups associated to a ramified leafwise flat line bundle on ariemannian foliation. Then we argue, by precise computations, that cohomologygroups associated to ramified leafwise flat vector bundles on a suitablefoliated space (whose existence is still unknown) might be useful to studyarithmetic L-functions via Atiyah-Bott-Lefschetz trace formulas.

On the analogy between L-functions and Atiyah-Bott-Lefschetz trace formulas for foliated spaces