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Schemes over 𝔽1 and zeta functions
- Part of
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- Journal:
- Compositio Mathematica / Volume 146 / Issue 6 / November 2010
- Published online by Cambridge University Press:
- 21 April 2010, pp. 1383-1415
- Print publication:
- November 2010
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- Article
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We determine the real counting function N(q) (q∈[1,∞)) for the hypothetical ‘curve’
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 
-schemes to interpret conceptually the spectral realization of zeros of L-functions.
over 𝔽
-schemes to interpret conceptually the spectral realization of zeros of 