Eisenstein group cocycles for GLn and values ofL-functions
Eisenstein group cocycles for GLn and values ofL-functions
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GLn 的爱森斯坦群余循环和 L 函数的值
DOI：
10.1007/bf01244319
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发表时间：
1993
影响因子：
3.1
通讯作者：
R. Sczech
中科院分区：
文献类型：
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作者：
R. Sczech
1.1设F是整数环上的n次全真实的数域，B，f是整数环中的两个互素理想~）r-射线类B mod f的部分zeta函数定义为~（B，f; s）=~ N（a）-% Re（s）> l a=--B（f）其中a遍历~ 3r中的所有整数理想使得分数次理想ab-~是由陪集1+ fb-t中的全正数生成的主理想。这个函数除了在s= 1处的一个简单极点外，对整个复平面都有解析延拓。它最初是由Hecke提出的，但首先由Siegel和Klingen证明，它在非正整数s处的特殊值是有理数：如果（B，f; 1-s）~ Q，s= 1，2，3.这一结果是补充Shintani谁建立了有限的公式，这些有理数的伯努利数，推广经典的欧拉公式：~（1-s）=-BJ s，s= 2，3，4.本文证明了么模群F= GL，f; 1-s的特殊值~（B，f; 1-s）可以用Eilenberg-MacLane群上同调来解释.我们构造了F上的一个群上圈（称为Eisenstein上圈），它表示H”-~（F，M）中的一个非平凡上同调类，其值在函数空间M中.限制ku到U，1+ f中的全正单位群（通过正则表示嵌入F），并在U-不变点上评估M的元素，产生H”-I（U，C）中具有平凡系数的上同调类。在这些类中，有一个有理上同调类序列q（B，f; s）eH”-1（U，~），s= 1，2.通过对H_I（U，Z）中的一个基本圈的求值，给出了数（（B，f; 1-s）. Eisenstein上循环7 '是普适的，因为它参数化了Hecke L-函数在每个n次全真实的数域中的所有特殊值，这些值已知是代数数或代数数乘以n的幂（参见[1]）。定理5）。
1.1 Let F be a totally real number field of degree n over~ and b, f two relatively prime ideals in the ring of integers~) r-The partial zeta function associated with the ray class b mod fis defined by~(b, f; s)=~ N (a)-% Re (s)> l a=--b (f) where a runs over all integral ideals in~ 3r such that the fractional ideal ab-~ is a principal ideal generated by a totally positive number in the coset 1+ fb-t. This function has an analytic continuation to the whole complex plane except for a simple pole at s= 1. It was originally conjectured by Hecke, but first proved by Siegel and Klingen, that its special values at non-positive integral s are rational numbers: if (b, f; 1-s)~ Q, s= 1, 2, 3..... This result was supplemented by Shintani who established a finite formula for these rational numbers in terms of Bernoulli numbers, generalizing the classical formula of Euler:~(1-s)=-BJs, s= 2, 3, 4.... In this paper, we show that the special values~(b, f; 1-s) admit a cohomological interpretation in terms of the Eilenberg-MacLane group cohomology of the unimodular group F= GL, 7/. We construct a group cocycle 7~ on F (called the Eisenstein cocycle) which represents a nontrivial cohomology class in H"-~(F, M) with values in a function space M. Restricting ku to U, the group of totally positive units in 1+ f (embedded in F via a regular representation), and evaluating the elements of M on U-invariant points, gives rise to cohomology classes in H"-I (U, C) with trivial coefficients. Among these classes, there is a sequence of rational cohomology classes q (b, f; s) e H"-1 (U,~), s= 1, 2..... which give the numbers ((b, f; 1-s) by evaluation on a fundamental cycle in H, _I (U, Z). The Eisenstein cocycle 7'is universal in the sense that it parameterizes all those special values of Hecke L-functions in every totally real number field of degree n which are known to be either an algebraic number or an algebraic number times a power of n (cf. Theorem 5).