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Schemes over \(\mathbb F_{1}\) and zeta functions. (English) Zbl 1201.14001
The authors determine the real counting function for the hypothetical “curve” \(C\) over \({\mathbb{F}}_1\), whose corresponding zeta function is the complete Riemann zeta function. Such a counting function exists as a distribution, is positive on \((1, \infty)\) and takes the value \(- \infty\) at \(q=1\) as expected from the infinite genus of \(C\).
As an application, the authors apply their functorial theory in order to interpret conceptually the spectral realization of zeros of L-functions.

14A15 Schemes and morphisms
14G10 Zeta functions and related questions in algebraic geometry (e.g., Birch-Swinnerton-Dyer conjecture)
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Full Text: DOI arXiv
[5] doi:10.3792/pjaa.82.141 · Zbl 1173.14004
[11] doi:10.1016/j.jnt.2008.08.007 · Zbl 1228.11143
[15] doi:10.1016/j.aim.2007.03.006 · Zbl 1125.14001
[16] doi:10.1007/s000290050042 · Zbl 0945.11015
[17] doi:10.1017/is008004027jkt048 · Zbl 1177.14022
[21] doi:10.1215/S0012-7094-04-12734-4 · Zbl 1079.11044
[22] doi:10.2307/2374941 · Zbl 0832.14002
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