Riemann-Roch Theorem, Stability and New Zeta Functions for Number Fields
Riemann-Roch Theorem, Stability and New Zeta Functions for Number Fields
复制标题
黎曼-罗赫定理、数域稳定性和新 Zeta 函数
DOI：
--
复制
发表时间：
2000
期刊：
影响因子：
--
通讯作者：
L. Weng
中科院分区：
文献类型：
--
作者：
L. Weng
本文引入了新的数域非阿贝尔zeta函数，并研究了它们的基本性质。回想一下，对于数字域，我们有经典的Dedekind zeta函数。这些函数通常被称为阿贝尔函数，因为在Artin之后，它们与伽罗瓦群的一维表示相关联；此外，继Tate和Iwasawa之后，它们可以构造为阿贝尔空间上的积分，即F在阿贝尔空间AF上的GL1积分。因此，为了定义数字域的非阿贝尔版本的ζ函数，数学家自然会使用伽罗瓦群和/或代数群的高维表示。这是非常重要和富有成效的。因此，现在我们有了所谓的马丁l函数，自同构l函数，等等。然而，在本文中，我们不打算触及这样一个迷人的表示为导向的数论理论的任何部分。相反，我们用更几何的方法来做。它包括两个方面，即积分域和集成域，以及Tata和Iwasawa的开创性作品。为了构造相当满意的积分，我们需要一个完备的上同理论，黎曼洛克定理的形式就是完备的上同理论。为此，在本文的第一部分中，对于具有KF是次为log |∆F |的正则元的数域F，我们首先引入了数域上向量束E的一个泛型；然后，我们定义了满足标准对偶性h(F，E) = h（F,E⊗KF）的向量束的第0上同调h（F，E）和第1上同调h(F，E)；最后证明了h(F，E) - h(F，E) = deg(E) - rank(E) 2·log |∆F |的黎曼-洛克定理。
In this paper, we introduce new non-abelian zeta functions for number fields and study their basic properties. Recall that for number fields, we have the classical Dedekind zeta functions. These functions are usually called abelian, since, following Artin, they are associated to one dimensional representations of Galois groups; moreover, following Tate and Iwasawa, they may be constructed as integrations over abelian spaces, i.e., GL1 over adelic space AF for F . Thus to define non-abelian versions of zeta functions for number fields, naturally, mathematicians use higher dimensional representations of Galois groups and/or algebraic groups. This turns to be extremely important and very fruitful. As a result, now we have the so-called Artin L-functions, automorphic Lfunctions, etc.. However in this paper, we are not going to touch any part of such a fascinating representation oriented number theoretical theory. Instead, we do it more geometrically. It consists of two aspects, i.e., the one for integrands and the one for integration domains, along with the pioneer works of Tata and Iwasawa. To construct quite satisfied integrands, we need a completed cohomology theory, form which RiemannRoch theorem holds. For this purpose, in Part I of this paper, for a number field F with KF a canonical element of degree log |∆F |, we first introduce an adelic version of vector bundles E over number fields; then, we define the 0-th cohomology h(F,E) and the 1-st cohomology h(F,E) for these vector bundles which satisfy the standard duality h(F,E) = h(F,E ⊗ KF ); and finally, we prove the following Riemann-Roch theorem for them: h(F,E) − h(F,E) = deg(E) − rank(E) 2 · log |∆F |.