Schemes over F1 and Zeta Functions

We determine the real counting function N(q) (q ∈ [1,∞)) for the hypothetical “curve” C = Spec Z over F1, whose corresponding zeta function is the complete Riemann zeta function. We show that such counting function exists as a distribution, is positive on (1,∞) and takes the value −∞ at q = 1 as expected from the infinite genus of C. Then, we develop a theory of functorial F1-schemes which reconciles the previous attempts by C. Soulé and A. Deitmar. Our construction fits with the geometry of monoids of K. Kato, is no longer limited to toric varieties and it covers the case of schemes associated to Chevalley groups. Finally we show, using the monoid of adèle classes over an arbitrary global field, how to apply our functorial theory of Mo-schemes to interpret conceptually the spectral realization of zeros of L-functions.