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SPECIAL VALUES OF THE ZETA FUNCTION OF AN ARITHMETIC SURFACE
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 21 / Issue 6 / November 2022
- Published online by Cambridge University Press:
- 15 March 2021, pp. 2043-2091
- Print publication:
- November 2022
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- Article
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- You have access
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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.
