Abstract
AbstractWe 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.
Highlights
Let X be a regular scheme of dimension d, proper over Spec(Z)
In [6, Conjectures 5.11 and 5.12], the first author and Morin formulated a conjecture on the vanishing order and the leading Taylor coefficient of the zeta function ζ(X,s) of X at integer arguments s = n ∈ Z in terms of what we call Weil–Arakelov cohomology complexes
In the present paper we focus on the case n = 1 and specialise further to arithmetic surfaces
Summary
Let X be a regular scheme of dimension d, proper over Spec(Z). In [6, Conjectures 5.11 and 5.12], the first author and Morin formulated a conjecture on the vanishing order and the leading Taylor coefficient of the zeta function ζ(X ,s) of X at integer arguments s = n ∈ Z in terms of what we call Weil–Arakelov cohomology complexes. This conjecture is necessary to define the Weil-étale complexes in terms of which our special value conjecture is formulated. RΓ X,IXi /IXi+1 is a perfect complex of Fp-modules, since X is proper It follows that the maps α and αadd are isomorphisms after tensoring with Qp. recall the theorem of formal functions [17, Theorem III.11.1]: RΓ(X,OX) ∼= RΓ XZp,OXZp. Isomorphism (12) is the composition of the scalar extensions to Qp of isomorphism (17), α, log, αadd and expression (19).
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