SCALING GROUP FLOW AND LEFSCHETZ TRACE FORMULA FOR LAMINATED SPACES WITH p−ADIC TRANSVERSAL

In his approach to analytic number theory C. Deninger has suggested that to the Riemann zeta function ζ̂(s) (resp. the zeta function ζY (s) of a smooth projective curve Y over a finite field Fq, q = p f )) one could possibly associate a foliated Riemannian laminated space (SQ,F , g, φ) (resp. (SY ,F , g, φ)) endowed with an action of a flow φ whose primitive compact orbits should correspond to the primes of Q (resp. Y ). Precise conjectures were stated in our report [Lei03] on Deninger’s work. The existence of such a foliated space and flow φ is still unknown except when Y is an elliptic curve (see Deninger [De02]). Being motivated by this latter case, we introduce a class of foliated laminated spaces (S = L×R +∗ q ,F , g, φ) where L is locally D × Zp , 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 φ acting on the leafwise Hodge cohomology H τ (0 ≤ j ≤ 2) of (S,F) that is very similar to the explicit formula for the zeta function of a (general) smooth curve over Fq. We also prove that the eigenvalues of the infinitesimal generator of the action of φ on H τ have real part equal to 1 2 . Moreover, we suggest in a precise way that the flow φ should be induced by a renormalization group flow ”à la K. Wilson”. We show that when Y is an elliptic curve over Fq this is indeed the case. It would be very interesting to establish a precise connection between our results and those of Connes (page 553 [Co00], page 90 [Co02]) and Connes-Marcolli [Co-Ma04a], [Co-Ma04b] on the Galois interpretation of the renormalization group.