Scaling group flow and Lefschetz trace formula for laminated spaces with p-adic transversal

Abstract

In his approach to analytic number theory C. Deninger has suggested that to the Riemann zeta function ζ ˆ ( s ) (resp. the zeta function ζ Y ( s ) of a smooth projective curve Y over a finite field F q , q = p f )) one could possibly associate a foliated Riemannian laminated space ( S Q , F , g , ϕ t ) (resp. ( S Y , F , g , ϕ t ) ) endowed with an action of a flow ϕ t whose primitive compact orbits should correspond to the primes of Q (resp. Y). Precise conjectures were stated in our report [E. Leichtnam, An invitation to Deninger's work on arithmetic zeta functions, in: Geometry, Spectral Theory, Groups, and Dynamics, in: Contemp. Math. vol. 387, Amer. Math. Soc., Providence, RI, 2005, pp. 201–236] on Deninger's work. The existence of such a foliated space and flow ϕ t is still unknown except when Y is an elliptic curve (see Deninger [C. Deninger, On the nature of explicit formulas in analytic number theory, a simple example, in: Number Theoretic Methods, Iizuka, 2001, in: Dev. Math., vol. 8, Kluwer Acad. Publ., Dordrecht, 2002, pp. 97–118]). Being motivated by this latter case, we introduce a class of foliated laminated spaces ( S = L × R + ∗ q Z , F , g , ϕ t ) where L is locally D × Z p m , 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 t acting on the leafwise Hodge cohomology H τ j ( 0 ⩽ j ⩽ 2 ) of ( S , F ) that is very similar to the explicit formula for the zeta function of a (general) smooth curve over F q . We also prove that the eigenvalues of the infinitesimal generator of the action of ϕ t on H τ 1 have real part equal to 1 2 . Moreover, we suggest in a precise way that the flow ϕ t should be induced by a renormalization group flow “à la K. Wilson”. We show that when Y is an elliptic curve over F q this is indeed the case. It would be very interesting to establish a precise connection between our results and those of Connes (p. 553 in [A. Connes, Noncommutative Geometry Year 2000, in: Special Volume GAFA 2000 Part II, pp. 481–559], p. 90 in [A. Connes, Symétries Galoisiennes et Renormalisation, in: Séminaire Bourbaphy, Octobre 2002, pp. 75–91]) and Connes–Marcolli [A. Connes, M. Marcolli, Q -lattices: quantum statistical mechanics and Galois theory, in: Frontiers in Number Theory, Physics and Geometry, vol. I, Springer-Verlag, 2006, pp. 269–350; A. Connes, M. Marcolli, From physics to number theory via noncommutative geometry. Part II: renormalization, the Riemann–Hilbert correspondence, and motivic Galois theory, in: Frontiers in Number Theory, Physics and Geometry, vol. II, Springer-Verlag, 2006, pp. 617–713] on the Galois interpretation of the renormalization group.