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SPECIAL VALUES OF THE ZETA FUNCTION OF AN ARITHMETIC SURFACE
Published online by Cambridge University Press: 15 March 2021
Abstract
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.
- Type
- Research Article
- Information
- Journal of the Institute of Mathematics of Jussieu , Volume 21 , Issue 6 , November 2022 , pp. 2043 - 2091
- Creative Commons
- 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
References
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