ON A CERTAIN SUM OF AUTOMORPHIC L-FUNCTIONS
ON A CERTAIN SUM OF AUTOMORPHIC L-FUNCTIONS
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关于一定数量的自同构 L 函数
DOI：
10.1090/conm/614/12270
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发表时间：
2013
影响因子：
1
通讯作者：
N. Chau
中科院分区：
文献类型：
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作者：
N. Chau
在Tate的论文[20]中，Zp的特征函数被用于与非分歧拟特征相关的局部L-因子的积分表示。Tamagawa，Godement和Jacquet在[19，8]中把这种构造推广到了GLn的主L-函数，并把整矩阵空间的特征函数作为检验函数。一般自守L-函数依赖于对偶群的一个表示。对于对偶群的每一个表示，我们可以构造一个依赖于复参数s的函数，其在非分歧表示上的迹是相关的局部L-因子。本文的主要贡献是通过Vinberg的代数幺半群理论[21]对这个测试函数进行几何描述。在我们之前，人们已经在这个方向上做出了努力，特别是Braverman和Kazhdan [4]和Lafforgue [11]。通过在迹公式的几乎每一个地方插入这些函数的乘积，我们将得到，至少在形式上，在[6，7]中考虑的L-函数的和。我们将解释这个L-函数和的几何解释。这篇论文是为了一个临时的目的，必要的细节和证明将在其他地方发表。我要感谢W。卡塞尔曼，D.江和裁判的评论，特别是Y。Sakellarlavia为启发性的意见交流。虽然我见过他一两次，但我并不有幸认识Piatetski-Shapiro教授。大约在2004年的某个时候，在卢米尼的一次会议上，他的妻子告诉我，Piatetski-Shapiro教授很欣赏我的作品。我把这句话作为我最自豪的数学奖之一，保留着美好的回忆。我把这部作品献给他，以表达对他的思想和勇气的深深钦佩。
In Tate’s thesis [20], the characteristic function of Zp has been used in the integral representation of the local L-factor associated to an unramified quasicharacter. This construction has been generalized by Tamagawa, Godement and Jacquet in [19, 8] to principal L-functions of GLn with the characteristic function of the space of integral matrices as test function. The general automorphic L-function depends on a representation of the dual group. For each representation of the dual group, one can construct a function depending on a complex parameter s whose trace on an unramified representation is the associated local L-factor. The main contribution of this paper is a conjectural geometric description of this test function by means of Vinberg’s theory of algebraic monoids [21]. Prior to us, efforts have been made in this direction, notably by Braverman and Kazhdan [4] and Lafforgue [11]. By inserting into the trace formula the product of these functions at almost every place, we would get, at least formally, the sum of L-functions considered in [6, 7]. We will explain a conjectural geometric interpretation of this sum of L-functions. This paper is intended for an expository purpose, necessary details and proofs will be published elsewhere. I would like to thank W. Casselman, D. Jiang and the referee for their comments, and especially Y. Sakellaridis for enlightening exchange of views. I wasn’t fortunate enough to know Professor Piatetski-Shapiro personally, though I met him once or twice. In a conference in Luminy some time around 2004, his wife told me that Professor Piatetski-Shapiro appreciated my works. I keep the fond memory of this comment as one of my proudest mathematical prizes. I dedicate this work to his memory as an expression of a deep admiration for his ideas and courage.