Schemes over 𝔽₁and zeta functions

Abstract

We determine the real counting function N (q) (q ∈ [1, ∞)) for the hypothetical 'curve' C = Spec Z over F 1 , whose corresponding zeta function is the complete Riemann zeta function. We show that such a 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 F 1 -schemes which reconciles the previous attempts by Soulé and Deitmar. Our construction fits with the geometry of monoids of Kato, is no longer limited to toric varieties and it covers the case of schemes associated with 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.

From monoids to hyperstructures: in search of an absolute arithmetic

“…By taking this into account, we obtain the following more precise result (for the proof we refer to [6], Theorem 2.2) Theorem 2.2. The tempered distribution N (u) satisfying the equation…”

“…it satisfies: 0x = x0 = 0, ∀x ∈ M . The theory of Mo-schemes as in our earlier papers [6] and [7] develops, in parallel with the classical theory of Z-schemes as in [14], the notion of a scheme as covariant functor from Mo to the category of sets. This approach is based on the earlier geometric theories over monoids developed by K. Kato [22], A. Deitmar [13], N. Kurokawa, H. Ochiai, M. Wakayama [26], B. Töen and M. Vaquié [33].…”

“…This result also prevents one to use the limit definition (1.1). In [6], we have shown how to solve these difficulties by replacing the limit definition (1.1) with an integral formula and by computing explicitly the distribution N (q) which fulfills the expected positivity, for q > 1, and the divergence at q = 1.…”

“…In particular, we overview several of our recent results (cf. [6]) showing that the natural monoidal structure on the adèle class space, when combined with one of the simplest examples of monoidal schemes i.e. the projective line P 1 F 1 , provides a geometric framework to understand conceptually the spectral realization of the zeros of L-functions, the functional equation and the explicit formulas.…”

Toward homological characterization of semirings by e-injective semimodules

“…For the readers' convenience, briefly remind some general notions for categories with zero morphisms (see, for example, [44,Chapters 7,8,13]) in the context of semimodule categories…”