Schemes over 𝔽1 and zeta functions
By Alain Connes, C. Consani. Compositio Mathematica, 2009
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Abstract We determine the real counting function N(q) (q∈[1,∞)) for the hypothetical ‘curve’ $C=\overline {\mathrm {Spec}\,\Z }$ over 𝔽1, 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 𝔽1-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.
Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics
View Paper: https://www.semanticscholar.org/paper/5e278ec0852c122c92e8690b27de480582af5d1b
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Citations in the corpus
- Mapping F_1-land:An overview of geometries over the field with one element (2009, background): “Schemes overF1 in the sense of Connes and Consani 10 1.7.”