Analytic Properties of Automorphic L-functions

自守 L-函数的解析性质

• 批准号: 0072450
• 负责人: I. Piatetski-Shapiro
• 金额: $ 11.57万
• 依托单位: Yale University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2000
• 资助国家: 美国
• 起止时间: 2000-07-15 至 2004-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0072450&HistoricalAwards=false
• 关键词: Analytic; Properties; Automorphic; functions

Abstract: Piatetski-Shapiro DMS-0072450The basic question described in the proposal is the problem of Langlands lifting. The definition of Langlands lifting can be done the following way. First one can define the local lifting. For an archimedean place one can define Langlands lifting in the following way. By Langlands theorem any irreducible admissible representation of a classical group one can attach a homomorphism of the Weil group to the L-group of this classical group. After that, for any classical group there is a natural embedding to the L-group of a general linear group. In this way one can attach a representation of the Weil group over a archimedean field a representation of the general linear group over that field. Similarly one can attach to any unramified representation of a classical group over a non-archimedean field we attach a semisimple conjugacy class of the L-group of the given classical group. The image of this conjugacy class in the corresponding general linear group L-group determines an unramified representation of the general linear group over the non-archimedean field. In this way to an automorphic representation of the classical group one can attach a collection of local representations of the general linear group defined at all archimedean places and at all unramified places. It is a well know statement that if a cuspidal automorphic representation of the general linear group exists with this data then it is unique. So the proposal is devoted to the study of Langlands lifting.Langlands lifting is a part of Langlands philosophy. I was asked by the program director for algebra and number theory to explain what the Langlands program is. I have thought about this a lot of time and I can say the following. Langlands philosophy gives a connection between automorphic forms and Dirichlet series. The most important example of a Dirichlet series is the famous Riemann zeta function. The automorphic form is the classical theta function of Jacobi. A second example the automorphic function is the so called Delta function which is a modular form of weight twelve. The corresponding Dirichlet series was studied first by Mordell who showed it had an Euler product. The coefficients of the Delta function are usually called the Ramanujan numbers. The basic properties of these numbers is that they are multiplicative. So the corresponding L-function has an Euler product and similar properties to the Riemann zeta function, namely holomorphic continuation the the whole complex plane and satisfies a functional equation. This connection can be extended to any algebraic reductive group. Let me explain it for the general linear group. Let's consider the space of functions which consists on all functions on the group of adeles invariant on the left side with respect to the group of principal adeles. Any smooth function of this type will be called an automorphic form. There is a way how one can attach to any automorphic form on the general linear group a Dirichlet series which has an Euler product and nice analytic properties, a nice functional equation, and holomorphic continuation to the whole complex plane.

摘要：Piatetski-Shapiro DMS-0072450提案中描述的基本问题是朗兰兹提升问题。朗兰兹提升的定义可以通过以下方式来完成。首先，我们可以定义局部提升。对于阿基米德场所，人们可以用以下方式定义朗兰兹提升。利用朗兰兹定理，一个经典群的任何不可约可容许表示都可以把Weil群的一个同态附加到这个经典群的L-群上。之后，对任一经典群，都有一个自然嵌入到一般线性群的L-群。这样，我们就可以在阿基米德域上附加Weil群的表示，在该域上附加一般线性群的表示。类似地，我们可以附加到非阿基米德域上的经典群的任何未分支表示上，我们附加了给定的经典群的L-群的一个半单共轭类。这一共轭类在相应的一般线性群L-群中的像决定了一般线性群在非阿基米德域上的一个非分支表示。通过这种方式，人们可以将定义在所有阿基米德位置和所有未分支位置的一般线性群的局部表示的集合附加到经典群的自同构表示。众所周知，如果具有该数据的一般线性群的尖角自同构表示存在，则它是唯一的。因此，该提案致力于朗兰兹提升的研究。朗兰兹提升是朗兰兹哲学的一部分。学习代数和数论的项目主管要求我解释朗兰兹项目是什么。我已经考虑了很长时间，我可以说以下几点。朗兰兹哲学给出了自同构形与狄里克莱级数之间的联系。狄里克莱级数最重要的例子是著名的Riemann Zeta函数。自同构形是雅可比的经典theta函数。自同构函数的第二个例子是所谓的增量函数，它是权12的模形式。相应的狄里克莱级数是由莫德尔首先研究的，他证明了狄里克莱级数存在欧拉积。Delta函数的系数通常称为Ramanujan数。这些数的基本性质是它们是乘法的。因此，对应的L函数具有欧拉积和类似于Riemann Zeta函数的性质，即在整个复平面上全纯连续，且满足一个函数方程。这种联系可以推广到任何代数约化群。让我来解释一下一般的线性群。让我们考虑函数空间，它由左侧的Adeles不变量群上的所有函数组成，相对于主Adeles群。这种类型的任何光滑函数都称为自同构形。有一种方法可以将Dirichlet级数附加到一般线性群上的任何自同构形上，该Dirichlet级数具有Euler积和良好的解析性质，良好的函数方程，并且在整个复平面上全纯连续。