Let \(\mathcal{X}\) be a regular connected proper scheme over \(Spec(\mathbb{Z})\) and \(\zeta(\mathcal{X},s)\) be its zeta function. The first named author of this paper and B. Morin, Doc. Math. 23, 1425–1560 (2018; Zbl 1404.14024)] made a conjecture on the vanishing order and the leading coefficient of the Taylor’s expansion at \(s=n\in \mathbb{Z}\). In the paper under review, the authors study the case \(n=1\) and \(\mathcal{X}\) is a proper regular flat arithmetic surface, and prove that this conjecture is equivalent to the Birch and Swinnerton-Dyer conjecture for the Jacobina of the generic fiber.
Indeed, the conjecture involved a rather inexplicit correction factor \(C(\mathcal{X},n)\in \mathbb{Q}^{\times}\), to ensure compatibility with the Tamagawa number conjecture of Bloch, Kato, Fontain, and Perrin-Riou. The first step of the proof of the equivalence is to show \(C(\mathcal{X},1)=1\) for arbitrary regular proper arithmetic schemes. For further study on this factor, refer to [M. Flach and B. Morin, Doc. Math. 26, 1633–1677 (2021; Zbl 1487.11065)].
The second step is a different proof of a formula due to Geisser relating the order of the Brauer and the Tate-Shafrevich group, where the authors remove the assumption on the base field to be totally imaginary in [T. H. Geisser, J. Inst. Math. Jussieu 19, No. 3, 965–970 (2020; Zbl 1474.11122)].
Finally, the study of the Neron modules gives the relations of the rest terms involved in these conjectural formulae, which needs the results of Bosch-Liu on components groups and Liu-Lorenzini-Raynaud on tangent spaces.