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

This paper is motivated by Deninger's programme. First we prove, using Alvarez Lopez-Kordyukov results, an Atiyah-Bott-Lefschetz trace formula for the cohomology groups associated to a ramified leafwise flat line bundle on a riemannian foliation. Then we argue, by precise computations, that cohomology groups associated to ramified leafwise flat vector bundles on a suitable foliated space (whose existence is still unknown) might be useful to study arithmetic L-functions via Atiyah-Bott-Lefschetz trace formulas.