Mathematics – Algebraic Geometry
Scientific paper
2009-03-11
Mathematics
Algebraic Geometry
1 figure, 32 pages
Scientific paper
We determine the {\em real} counting function $N(q)$ ($q\in [1,\infty)$) for the hypothetical "curve" $C=\overline {\Sp \Z}$ over $\F_1$, whose corresponding zeta function is the complete Riemann zeta function. Then, we develop a theory of functorial $\F_1$-schemes which reconciles the previous attempts by C. Soul\'e 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 ad\`ele 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.
Connes Alain
Consani Caterina
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