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• W3098984909 abstract "Abstract We determine the real counting function N ( q ) ( q ∈[1, ∞ )) for the hypothetical ‘curve’ $C=overline {mathrm {Spec},Z }$ 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." @default.
• W3098984909 created "2020-11-23" @default.
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• W3098984909 date "2010-04-21" @default.
• W3098984909 modified "2023-10-07" @default.
• W3098984909 title "Schemes over 𝔽₁and zeta functions" @default.
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• W3098984909 doi "https://doi.org/10.1112/s0010437x09004692" @default.
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