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Schemes over 𝔽1 and zeta functions

Part of:Arithmetic problems. Diophantine geometryArithmetic algebraic geometryAlgebraic geometry: Foundations

Published online by Cambridge University Press:  21 April 2010

Alain Connes and
Caterina Consani
Alain Connes
Affiliation:
Collège de France 3, I.H.E.S. and Vanderbilt University, rue d’Ulm, Paris F-75005, France (email: alain@connes.org)
Caterina Consani
Affiliation:
Mathematics Department, The Johns Hopkins University, Baltimore, MD 21218, USA (email: kc@math.jhu.edu)
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Abstract

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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.

Keywords

absolute pointcounting functionszeta functions over 𝔽1𝔽1-schemesprojective adèle class spacespectral realization of zeros of L-functions

MSC classification

Secondary:14A15: Schemes and morphisms14G10: Zeta-functions and related questions (Birch-Swinnerton-Dyer conjecture)11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture

Information

Type
Research Article
Information
Compositio Mathematica , Volume 146 , Issue 6 , November 2010, pp. 1383 - 1415
Copyright
Copyright © Foundation Compositio Mathematica 2010

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