Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves?
Is There an Analytic Theory of Automorphic Functions for Complex Algebraic Curves?
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是否存在复杂代数曲线自守函数的解析理论？
DOI：
10.3842/sigma.2020.042
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发表时间：
2018
影响因子：
0.9
通讯作者：
E. Frenkel
中科院分区：
文献类型：
--
作者：
E. Frenkel
复代数曲线的几何朗兰兹对应不同于数域的原始朗兰兹对应，因为它是用层而不是函数来表示的（在有限域上的曲线的中间情况下，两种公式都是可能的）。在最近的预印本，罗伯特朗兰兹提出了一个建议，发展分析理论的自守形式的模空间的$G$-丛的一个复杂的代数曲线。朗兰兹设想这些形式作为Hecke算子的某些类似物的特征函数。在这些说明中，我表明，如果$G$是一个阿贝尔群，那么有定义良好的赫克运营商，我给一个完整的描述，他们的本征函数和本征值。对于非阿贝尔$G$，赫克运营商涉及整合，这提出了一些困难。然而，有另一种方法来发展分析理论的自守形式，基于存在一个大的交换代数的全球微分算子作用于半密度上的模堆叠的$G$-丛。这里概述了这种方法（实现了Joerg Teschner的一些想法），作为与Pavel Etingof和大卫Kazhdan联合工作的预览。
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 abelian group then there are well-defined Hecke operators, and I give a complete description of their eigenfunctions and eigenvalues. For non-abelian $G$, Hecke operators involve integration, which presents some difficulties. However, there is an alternative approach to developing an analytic theory of automorphic forms, based on the existence of a large commutative algebra of global differential operators acting on half-densities on the moduli stack of $G$-bundles. This approach (which implements some ideas of Joerg Teschner) is outlined here, as a preview of a joint work with Pavel Etingof and David Kazhdan.
评估字段与本地字段的集成
影响因子：
0.6
作者：
Morrow Matthew
通讯作者：
Morrow Matthew