论文标题
托里超过数字字段和s = 1的特殊值
Tori over number fields and special values at s=1
论文作者
论文摘要
我们定义了一个weil-étale复合体，并对双重循环的双重支持（在Bloch双重化的意义上）进行了$ \ mathbb {z}^c $的$ \ Mathbb {Z} $ - 在$ 1 $ 1 $ -Diperal Arithmentional Arithmentimal Arithment flat flat flat flat flat conply ymath $ seppect y mathrm上}的$ \ s rmm {bmmath）$ \ ymmmmmm {c）可以将这种复合物视为计算Weil-étale同源性。对于那些$ \ mathbb {z} $ - 被驯服的构造式带状绳索，我们定义了一个“添加剂”复合物，我们认为它是$ \ mathbb {z} $的双重代数 - 可构造的融资。添加剂和Weil-étale综合体的决定因素的乘积称为基本线。我们证明了二元定理，这意味着基本线具有自然的琐碎化，从而赋予了倍增的欧拉特征。我们将天然$ l $ - 功能附加到$ \ mathbb {z} $的双重二元上 - 可构造的捆；最多有有限的因素，此$ l $ function是$ s+1 $的artin $ l $ function。我们的主要定理包含$ l $ unction的消失订单公式，$ s = 0 $，并指出，在驯服的情况下，$ s = 0 $的特殊值由Euler特征给出以签名。这概括了Dedekind Zeta函数的特殊值的分析类号码公式。在函数字段中，这是ARXIV的定理：2009.14504。
We define a Weil-étale complex with compact support for duals (in the sense of the Bloch dualizing cycles complex $\mathbb{Z}^c$) of a large class of $\mathbb{Z}$-constructible sheaves on an integral $1$-dimensional proper arithmetic scheme flat over $\mathrm{Spec}(\mathbb{Z})$. This complex can be thought of as computing Weil-étale homology. For those $\mathbb{Z}$-constructible sheaves that are moreover tamely ramified, we define an "additive" complex which we think of as the Lie algebra of the dual of the $\mathbb{Z}$-constructible sheaf. The product of the determinants of the additive and Weil-étale complex is called the fundamental line. We prove a duality theorem which implies that the fundamental line has a natural trivialization, giving a multiplicative Euler characteristic. We attach a natural $L$-function to the dual of a $\mathbb{Z}$-constructible sheaf; up to a finite number of factors, this $L$-function is an Artin $L$-function at $s+1$. Our main theorem contains a vanishing order formula at $s=0$ for the $L$-function and states that, in the tamely ramified case, the special value at $s=0$ is given up to sign by the Euler characteristic. This generalizes the analytic class number formula for the special value at $s=1$ of the Dedekind zeta function. In the function field case, this a theorem of arXiv:2009.14504.
