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SPECIAL VALUES OF THE ZETA FUNCTION OF AN ARITHMETIC SURFACE

Part of: Arithmetic algebraic geometry (Co)homology theory

Published online by Cambridge University Press:  15 March 2021

Matthias Flach Open the ORCID record for Matthias Flach [Opens in a new window]  and
Daniel Siebel
Matthias Flach
Affiliation:
Dept. of Mathematics 253-37, California Institute of Technology, PasadenaCA91125 (flach@caltech.edu, daniel.siebel@yahoo.de)
Daniel Siebel
Affiliation:
Dept. of Mathematics 253-37, California Institute of Technology, PasadenaCA91125 (flach@caltech.edu, daniel.siebel@yahoo.de)
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Abstract

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We prove that the special-value conjecture for the zeta function of a proper, regular, flat arithmetic surface formulated in [6] at $s=1$ is equivalent to the Birch and Swinnerton-Dyer conjecture for the Jacobian of the generic fibre. There are two key results in the proof. The first is the triviality of the correction factor of [6, Conjecture 5.12], which we show for arbitrary regular proper arithmetic schemes. In the proof we need to develop some results for the eh-topology on schemes over finite fields which might be of independent interest. The second result is a different proof of a formula due to Geisser, relating the cardinalities of the Brauer and the Tate–Shafarevich group, which applies to arbitrary rather than only totally imaginary base fields.

Keywords

arithmetic surfaceszeta valuesWeil-étale cohomologyBirch and Swinnerton-Dyer conjecture

MSC classification

Primary: 14F20: Étale and other Grothendieck topologies and (co)homologies
Secondary: 14F42: Motivic cohomology; motivic homotopy theory 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 6 , November 2022 , pp. 2043 - 2091
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Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2021. Published by Cambridge University Press

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