自守形式的表示和估计的解析延拓

自守分布的Rankin-Selberg方法

The Rankin-Selberg method for automorphic distributions

本文介绍了我们的方法配对自守分布。这代表了第三种技术获得的分析性质的自守L-函数，除了现有的方法的整表示（兰金-塞尔伯格）和傅里叶系数的爱森斯坦系列（朗兰兹-沙希迪）。最近，我们使用这种技术，以建立新的情况下，充分解析延拓的外部平方L-函数。本文在两种特殊而又有代表性的情况下，即PGL（2，Z）的Rankin-Selberg张量积L-函数和GL（4，Z）的外平方L-函数，阐述了我们的方法。

GL的自守分布、L函数和Voronoi求和（3）

Automorphic Distributions, L-functions, and Voronoi Summation for GL(3)

本文是继“求和公式，从泊松和Voronoi到现在”（数学NT/0304187）和“狄利克雷级数的分布和解析延拓“（数学FA/0403030）之后的第三篇。第一个主要是一个简要的文件，解释目前的一个，而第二个包含一些分配机制，以及在这里使用。这些文件关注的边界分布的自守形式，以及如何将它们应用于研究问题的尖点形式上半单李群。本文的主要结果是GL（3，Z）\GL（3，R）上尖点形式傅里叶系数的Voronoi型求和公式。我们还给出了GL（3）上的标准L-函数的处理，重点是使用分布进行的阿基米德分析。最后给出了Jacquet，Piatetski-Shapiro和Shalika的GL（3）xGL（1）逆定理的一个新的证明.本文也与后来的数学论文math.NT/0402382和math.NT/0404521有关。

Norm computation and analytic continuation of vector valued holomorphic discrete series representations

In this paper we compute explicitly the norm of the vector-valued holomorphic discrete series representations, when its $K$-type is "almost multiplicity-free." As an application, we discuss the properties of highest weight modules, such as unitarizability, reducibility and composition series.

Analytic continuation of equivariant distributions

We establish a method for constructing equivariant distributions on smooth real algebraic varieties from equivariant distributions on Zariski open subsets. This is based on Bernstein's theory of analytic continuation of holonomic distributions. We use this to construct $H$-equivariant functionals on principal series representations of $G$, where $G$ is a real reductive group and $H$ is an algebraic subgroup. We also deduce the existence of generalized Whittaker models for degenerate principal series representations. As a special case, this gives short proofs of existence of Whittaker models on…

Automorphic pairs of distributions on $\mathbb{R}$, and Maass forms of real weights

We give a correspondence between automorphic pairs of distributions on $\mathbb{R}$ and Dirichlet series satisfying functional equations and some additional analytic conditions. Moreover, we show that the notion of automorphic pairs of distributions on $\mathbb{R}$ can be regarded as a generalization of automorphic distributions on smooth principal series representations of the universal covering group of $SL(2,\mathbb{R})$. As an application, we prove Weil type converse theorems for automorphic distributions and Maass forms of real weights.

The Exterior Cubic L-function of GU(6) and Unitary Automorphic Induction

In this paper, we extend Ginzburg-Rallis' integral representation for the exterior cube automorphic $L$-function of ${\rm GL}_6\times {\rm GL}_1$ to that of the quasi-split unitary similitude group ${\rm GU}_6$ and establish its analytic properties to determine the poles of this $L$-function. Furthermore, we introduce the automorphic induction for automorphic representations of ${\rm GU}_n$ and then show that the weak Langlands functorial lift for the automorphic induction exists for generic cuspidal automorphic representations. By using this automorphic induction, we give a conjectural criter…

Galois representations for even general special orthogonal groups

We prove the existence of $\mathrm{GSpin}_{2n}$-valued Galois representations corresponding to cohomological cuspidal automorphic representations of certain quasi-split forms of $\mathrm{GSO}_{2n}$ under the local hypotheses that there is a Steinberg component and that the archimedean parameters are regular for the standard representation. This is based on the cohomology of Shimura varieties of abelian type, of type $D^{\mathbb{H}}$, arising from forms of $\mathrm{GSO}_{2n}$. As an application, under similar hypotheses, we compute automorphic multiplicities, prove meromorphic continuation of (…

Weakly holomorphic modular forms and rank two hyperbolic Kac-Moody algebras

In this paper, we compute basis elements of certain spaces of weight 0 weakly holomorphic modular forms and consider the integrality of Fourier coefficients of the modular forms. We use the results to construct automorphic correction of the rank 2 hyperbolic Kac-Moody algebras H(a), a=4,5,6, through Hilbert modular forms explicitly given by Borcherds lifts of the weakly holomorphic modular forms. We also compute asymptotic of the Fourier coefficients as they are related to root multiplicities of the rank 2 hyperbolic Kac-Moody algebras. This work is a continuation of the previous paper, where …

Is there an analytic theory of automorphic functions for complex algebraic curves?

The geometric Langlands correspondence for complex algebraic curves differs from the original Langlands correspondence for number fields in that it is formulated in terms of sheaves rather than functions (in the intermediate case of curves over finite fields, both formulations are possible). In a recent preprint, Robert Langlands made a proposal for developing an analytic theory of automorphic forms on the moduli space of $G$-bundles on a complex algebraic curve. Langlands envisioned these forms as eigenfunctions of some analogues of Hecke operators. In these notes I show that if $G$ is an abe…

Weights, raising and lowering operators, and K-types for automorphic forms on SL(3,R)

We give a fully explicit description of Lie algebra derivatives (generalizing raising and lowering operators) for representations of SL(3,R) in terms of a basis of Wigner functions. This basis is natural from the point of view of principal series representations, as well as computations in the analytic theory of automorphic forms (e.g., with Whittaker functions). The method is based on the Clebsch-Gordan multiplication rule for Wigner functions, and applies to other Lie groups whose maximal compact subgroup is isogenous to a product of SU(2) and U(1) factors. As an application, we give a compl…

代数环面的Weyl律

The Weyl law for algebraic tori

本文利用局部Langlands对应定义代数环面的解析导子，给出有界解析导子下自守特征数量的渐近估计，并将其纳入约化群自守表示计数的一般框架。

关于 Franke 定理的最简单情形

On Franke's theorem in the simplest case

本文直接证明上半平面上的球面自守形式可分解为尖点形式与标准 Eisenstein 级数 Laurent 系数的线性组合，这是 Franke 一般定理的最简单实例，证明基于自守形式的基本解析性质和 Green 恒等式。

The orbit method and analysis of automorphic forms

We develop the orbit method in a quantitative form, along the lines of microlocal analysis, and apply it to the analytic theory of automorphic forms. Our main global application is an asymptotic formula for averages of Gan--Gross--Prasad periods in arbitrary rank. The automorphic form on the larger group is held fixed, while that on the smaller group varies over a family of size roughly the fourth root of the conductors of the corresponding $L$-functions. Ratner's results on measure classification provide an important input to the proof. Our local results include asymptotic expansions for cert…