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 $\hat{\zeta}(s)$ (resp. the zeta function $\zeta_Y(s)$ of a smooth projective curve $Y$ over a finite field $\mathbb{F}_q$, $q=p^f$)) one could possibly associate a foliated Riemannian laminated space $(S_{\mathbb{Q}}, \mathcal{F}, g, \phi^t)$ (resp. $(S_Y, \mathcal{F}, g, \phi^t)$) endowed with an action of a flow $\phi^t$ whose primitive compact orbits should correspond to the primes of $\mathbb{Q}$ (resp. $Y$). The existence of such a foliated space and flow $\phi^t$ 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 $(S=\frac{\mathcal{L}\times \R^{+*}}{q^\Z}, \mathcal{F}, g, \phi^t)$ where $\mathcal{L} $ is locally $D\times \Z_p^m$, $D$ being an open disk of $\mathbb{C}.$ Assuming that the leafwise harmonic forms on $ \mathcal{L}$ are locally constant transversally, we prove a Lefschetz trace formula for the flow $\phi^t$ acting on the leafwise Hodge cohomology $H^j_\tau$ ($0\leq j \leq 2$) of $(S,\mathcal{F})$ that is very similar to the explicit formula for the zeta function of a (general) smooth curve over $\mathbb{F}_q$. We also prove that the eigenvalues of the infinitesimal generator of the action of $\phi^t$ on $H^1_\tau$ have real part equal to ${1/2}.$ Moreover, we suggest in a precise way that the flow $\phi^t$ should be induced by a renormalization group flow "\`a la K. Wilson". We show that when $Y$ is an elliptic curve over $\mathbb{F}_q$ this is indeed the case.