ε-factor of a tamely ramified sheaf on a variety
ε-factor of a tamely ramified sheaf on a variety
复制标题
品种上驯服分枝束的 ε 因子
DOI：
10.1007/bf01244312
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发表时间：
1993
影响因子：
3.1
通讯作者：
Takeshi Saito
中科院分区：
文献类型：
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作者：
Takeshi Saito
对于有限域上的簇上的平滑 (-adic 束)，S. Saito [SS] 证明了 e 因子的公式（L 函数的函数方程的常数项）。根据他的公式，它等于在簇的正则循环上评估的相应表示的行列式特征的值。在本文中，我们将其推广到驯服的分支束，并证明了局部域上的簇的类似公式。在我们的公式中，正则循环被一个精化的对象所取代，即在具有紧支持的上同调中定义的相对正则循环，并且出现了局部域上的变体是布洛赫[B]和加藤[K1]的导体公式的算术版本。 令X是特征p的完美域F上的维度n的适当平滑变体，并且令U是X的开，使得补数D是具有简单法线交叉的X的除数。相对正则循环 Cx. v 在紧支持 2, 7f'Hc (U, 77'(n)) for=//q Zq 中定义。它是第 1 节中定义的阶 n 的部分平凡化局部自由束 O~/F (logD) 的相对顶级陈类。如果一个族满射 8 l D,-}(gD，在满足某个性质的闭子方案 Di 上给出。至于 O~/v (logD)，D 的不可约分量 Di 处的留数映射族提供了部分平凡化。G. Anderson 发现应该存在部分平凡化局部自由束的相对陈类的良好定义。假设 F 是有限的，并令： 是质数~ p 且~ 是光滑的(-U 上的 adic 束。然后根据 Grothendieck，L 函数满足以下公式
For a smooth (-adic sheaf on a variety over a finite field, a formula for the e-factor (the constant term of the functional equation of the L-function) is proved by S. Saito [SS]. According to his formula, it equals the value of the determinant character of the corresponding representation evaluated at the canonical cycle of the variety. In this paper, we generalize this to a tamely ramified sheaf and also prove an analogous formula for a variety over a local field. In our formula, the canonical cycle is replaced by a refined object, the relative canonical cycle defined in the cohomology with compact support, and a certain product of Gauss sums appears. That for a variety over a local field is an arithmetic version of the conductor formula of Bloch [B] and that of Kato [K1]. Let X be a proper smooth variety of dimension n over a perfect field F of characteristic p and let U be an open of X such that the complement D is a divisor of X with simple normal crossings. Then the relative canonical cycle Cx. v is defined in the cohomology with compact support 2, 7f'Hc (U, 77'(n)) for=//q. p Zq. It is (-1)"-times of the relative top chern class of the partially trivialized locally free sheaf O~/F (logD) of rank n defined in Sect. 1. A locally free sheaf~ is said to be partially trivialized if a family of surjections 8 l D,-}(gD, on closed subschemes Di satisfying a certain property is given. As for O~/v (logD), the family of the residue maps at the irreducible components Di of D provides the partial trivialization. The observation that there should exist a good definition of the relative chern class of a partially trivialized locally free sheaf is due to G. Anderson. Assume F is finite and let: be a prime number~ p and~ be a smooth (-adic sheaf on U. Then by Grothendieck, the L-function satisfies the formula