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.

我们证明了在 $s=1$ 处在 [6] 中公式化的适当的、规则的、平坦的算术曲面的 zeta 函数的特殊值猜想等同于通用纤维的 Jacobian 的 Birch 和 Swinnerton-Dyer 猜想。证明中有两个关键结果。第一个是 [6，猜想 5.12] 的校正因子的微不足道，我们为任意正则本真算术方案展示了这一点。在证明中，我们需要为有限域上的格式的 eh 拓扑开发一些结果，这可能是独立的兴趣。第二个结果是 Geisser 对公式的不同证明，将 Brauer 和 Tate-Shafarevich 群的基数联系起来，它适用于任意基场，而不仅仅是完全虚构的基场。