Schemes over F1 and Zeta Functions

We develop a theory of schemes over the field of characteristic one which reconciles the previous attempts by Soulé and by Deitmar. Our construction fits with the geometry of monoids of Kato and is no longer limited to toric varieties. We compute the zeta function of an arbitrary Noetherian scheme (over the field of characteristic one) and prove that the torsion in the local geometric structure introduces ramification. Then we show that Soulé's definition of the zeta function of an algebraic variety over F 1 is equivalent to an integral formula. This result provides one with a way to extend the definition of such a function to the case of an arbitrary counting function with polynomial growth. We test this construction on elliptic curves over the rational numbers. Finally, we compare the above mentioned integral formula with the explicit formulae of number theory and we determine the counting function for the hypothetical curve Spec Z over the field of characteristic one.