2006/03/24 by Éric Leichtnam, Eric Leichtnam, Leichtnam, Eric
Mathematics · #Advanced Algebra and Geometry #Algebraic Geometry and Number Theory #Analytic Number Theory Research #Differential Geometry (math.DG) #FOS: Mathematics #Number Theory (math.NT) #Operator Algebras (math.OA) #math.DG #math.NT #math.OA
paper · pdf · doi:10.48550/arxiv.math/0603576
27 pages; v2: typos have been corrected
openalex publication_date 2006/03/24 · arxiv created 2006/05/02 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In his approach to analytic number theory C. Deninger has suggested that to the Riemann zeta function ζ(s) (resp. the zeta function ζY(s) of a smooth projective curve Y over a finite field \mathbbFq, q=pf)) one could possibly associate a foliated Riemannian laminated space (Sℚ, F, g, ϕt) (resp. (SY, F, g, ϕt)) endowed with an action of a flow ϕt whose primitive compact orbits should correspond to the primes of ℚ (resp. Y). The existence of such a foliated space and flow ϕt is still unknown except when Y is an elliptic curve (see Deninger). Being motivated by this latter case, we introduce a class of foliated laminated spaces (S=\fracL× \R+*q^\Z, F, g, ϕt) where L is locally D× \Zpm, D being an open disk of ℂ. Assuming that the leafwise harmonic forms on L are locally constant transversally, we prove a Lefschetz trace formula for the flow ϕt acting on the leafwise Hodge cohomology Hjτ (0≤ j ≤ 2) of (S,F) that is very similar to the explicit formula for the zeta function of a (general) smooth curve over \mathbbFq. We also prove that the eigenvalues of the infinitesimal generator of the action of ϕt on H1τ have real part equal to 1/2. Moreover, we suggest in a precise way that the flow ϕt should be induced by a renormalization group flow "à la K. Wilson". We show that when Y is an elliptic curve over \mathbbFq this is indeed the case.