Non-abelian zeta functions for function fields
Non-abelian zeta functions for function fields
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DOI：
10.1353/ajm.2005.0035
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发表时间：
2005-10
影响因子：
1.7
通讯作者：
L. Weng
中科院分区：
文献类型：
--
作者：
L. Weng
在本文中，我们针对有限域上定义的曲线发起了非交换 zeta 函数的几何定向构造。更准确地说，我们首先通过使用这些点的模解释对半稳定向量丛的相应模空间上的有理点进行“加权计数”，为在有限域上定义的曲线引入新的真正的非交换zeta函数。然后，我们使用与某些广义模空间上的 L2 自同构形式相关的艾森斯坦级数的积分，定义有限域上曲线的非阿贝尔 L 函数。介绍。在本文中，我们针对有限域上定义的曲线发起了非交换 zeta 函数的几何定向构造。它由两章组成。更准确地说，在第一章中，我们首先为有限域上定义的曲线引入新的、真正的非交换 zeta 函数。这是通过使用这些点的模解释对半稳定向量丛的相应模空间上的有理点进行“加权计数”来实现的。我们通过建立这些新 zeta 的基本属性（例如函数方程和理性）来证明我们的构造是正确的，并表明如果仅涉及线丛，我们新定义的 zeta 与 Artin 的 Zeta 一致。所有这些，特别是理性，自然地引出了我们对（全局）非阿贝尔 zeta 函数（对于在数域上定义的曲线）的定义，其本身由收敛结果证明是合理的。我们以对亏格二曲线的二阶非交换 zeta 函数的详细研究作为本章的结尾，该函数基于我们所说的 Brill-Noether 轨迹（和 Weierstrass 点）的无穷小结构。在第二章中，我们从理性领域的类似构造开始，以激发接下来的内容。特别是，我们证明了我们的非阿贝尔 zeta 函数和爱森斯坦级数之间存在内在联系。因此，我们不再为有限域上定义的曲线引入更通用的测试函数的一般非交换 L 函数（就像 Tate 在他的关于交换 L 函数的论文中所做的那样），而是将有限域上的曲线的非交换 L 函数定义为与某些广义模空间上的 L2 自守形式相关的爱森斯坦级数的积分。在这里，几何截断起着关键作用。这些非阿贝尔 L 函数的基本性质，例如亚纯延拓，
In this paper we initiate a geometrically oriented construction of non-abelian zeta functions for curves defined over finite fields. More precisely, we first introduce new yet genuine non-abelian zeta functions for curves defined over finite fields, by a "weighted count" on rational points over the corresponding moduli spaces of semi-stable vector bundles using moduli interpretation of these points. Then we define non-abelian L-functions for curves over finite fields using integrations of Eisenstein series associated to L2-automorphic forms over certain generalized moduli spaces. Introduction. In this paper we initiate a geometrically oriented construction of non-abelian zeta functions for curves defined over finite fields. It consists of two chapters. More precisely, in Chapter I, we first introduce new yet genuine non-abelian zeta functions for curves defined over finite fields. This is achieved by a "weighted count" on rational points over the corresponding moduli spaces of semi-stable vector bundles using moduli interpretation of these points. We justify our con- struction by establishing basic properties for these new zetas such as functional equation and rationality, and show that if only line bundles are involved, our newly defined zetas coincide with Artin's Zeta. All this, in particular, the ratio- nality, then leads naturally to our definition of (global) non-abelian zeta functions (for curves defined over number fields), which themselves are justified by a con- vergence result. We end this chapter with a detailed study on rank two non-abelian zeta functions for genus two curves, based on what we call infinitesimal structures of Brill-Noether loci (and Weierstrass points). In Chapter II, we begin with a similar construction for the field of rationals to motivate what follows. In particular, we show that there is an intrinsic relation between our non-abelian zeta functions and Eisenstein series. Due to this, instead of introducing general non-abelian L-functions for curves defined over finite fields with more general test functions (as what Tate did in his Thesis for abelian L- functions), we then define non-abelian L-functions for curves over finite fields as integrations of Eisenstein series associated to L2-automorphic forms over certain generalized moduli spaces. Here geometric truncations play a key role. Basic properties for these non-abelian L-functions, such as meromorphic continuation,