2019/09/16 by Matthias Flach, Daniel Siebel · 4 citations
Computer Science · Mathematics · #Algebraic Geometry and Number Theory #Arithmetic zeta function #Base (topology) #Coding theory and cryptography #Conjecture #Function (biology) #Polynomial and algebraic computation #Riemann zeta function #Surface (topology) #Triviality #math.NT #msc:11G40 #msc:14F10 #msc:14F20
paper · pdf · doi:10.1017/s1474748021000104
published in Journal of the Institute of Mathematics of Jussieu 21(6), 2043-2091 (Cambridge University Press)
arxiv created 2019/09/16 · openalex created_date 2019/09/26 · openalex publication_date 2021/03/15 · arxiv updated 2021/07/01 · openalex updated_date 2026/08/05
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.