权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Explicit Geometric Langlands Correspondence for Rigid Local Systems

刚性局部系统的显式几何朗兰兹对应

基本信息

批准号：
418779201
负责人：
Dr. Konstantin Jakob
金额：
--
依托单位：
Department of Mathematics
依托单位国家：
德国
项目类别：
Research Fellowships
财政年份：
2019
资助国家：
德国
起止时间：
2018-12-31 至 2021-12-31
项目状态：
已结题

来源：
https://gepris.dfg.de/gepris/projekt/418779201?language=en
关键词：
Explicit Geometric Langlands Correspondence Rigid

项目摘要

Probably the best known example of a rigid local system is the solution sheaf of the Gaussian hypergeometric equation on the complex-projective line P^1 punctured at 0, 1 and infinity. In this context, the rigidity condition says that this solution sheaf (which is a local system) is determined up to isomorphism by the local monodromy around the punctures. This local monodromy at a point is given as a linear relation between a fundamental solution of the differential equation and its analytic continuation along a simple path around the missing point. The Gaussian equation has the special property that all its singular points are regular singular. Roughly speaking this means that the solutions are subject to some growth condition. Rigid local systems arising from regular singular differential equations in this way have found application in inverse Galois theory and the construction of motives with exceptional motivic Galois groups. More generally one can define a similar notion for differential equations resp. connections on trivial vector bundles which are not necessarily regular singular. In this case such a connection is called rigid if its isomorphism class is determined uniquely by local formal data. There are essentially two ways to construct rigid connections resp. in the setting of a finite base field, l-adic local systems. The first of these is a theorem of Katz & Arinkin which states that any irreducible rigid connection on an open subset of P^1 can be constructed from a connection of rank one by iterating Fourier-Laplace transform, twisting with a connection of rank one or coordinate changes by a Möbius transform. In the regular singular (resp. tamely ramified) case, one can replace Fourier-Laplace transform by middle convolution defined by Katz. This method was used by Dettweiler & Reiter to classify tamely ramified rigid local systems with monodromy group the simple exceptional algebraic group of type G_2. Additionally I used this method to construct new rigid irregular connections with differential Galois group of type G_2. The second way of constructing rigid local systems is the following. Heinloth, Ngô & Yun use the geometric Langlands correspondence to construct local systems as eigensystems of Hecke-eigensheaves on the moduli space of G-bundles with level structure. They constructed Kloosterman sheaves for reductive groups, realizing several exceptional algebraic groups as geometric monodromy groups.The aim of this project is the generalization of Heinloth, Ngô & Yun’s construction to obtain new classes of rigid local systems. Additionally we wish to reobtain certain known examples. In particular we hope to obtain an automorphic interpretation of Katz’s hypergeometric sheaves. These are l-adic analogues and generalizations of the Gaussian hypergeometric equation mentioned above.

也许最著名的刚性局部系统的例子是高斯超几何方程在复射影线P^1上的解束，P^1在0,1和无穷远处刺穿。在这种情况下，刚性条件表明，该解束（这是一个局部系统）是由穿孔周围的局部单调决定的，直至同构。这一点上的局部单一性被表示为微分方程的基本解和它的解析延拓沿绕缺失点的简单路径之间的线性关系。高斯方程有一个特殊的性质，即它的所有奇异点都是正则奇异。粗略地说，这意味着解决方案受制于某些增长条件。用这种方法由正则奇异微分方程产生的刚性局部系统在逆伽罗瓦理论和特殊伽罗瓦群动机的构造中得到了应用。更一般地说，我们可以为微分方程定义一个类似的概念。平凡向量束上不一定是正则奇异的连接。在这种情况下，如果这种连接的同构类是由局部形式数据唯一确定的，则称为刚性连接。基本上有两种构建刚性连接的方法。在有限基域的设置下，l进进局部系统。第一个是Katz & Arinkin的定理，该定理指出在P^1的开放子集上任何不可约的刚性连接都可以通过傅里叶-拉普拉斯变换的迭代，用一级连接的扭转或通过Möbius变换的坐标变化从一级连接中构造出来。用规则的单数形式(如：在纯分支情况下，可以用卡茨定义的中卷积代替傅里叶-拉普拉斯变换。Dettweiler & Reiter用这种方法对具有单群的组分枝刚性局部系统（G_2型的简单例外代数群）进行了分类。此外，还利用该方法构造了G_2型微分伽罗瓦群的刚性不规则连接。第二种构造刚性局部系统的方法如下。Heinloth， Ngô & Yun利用几何Langlands对应在具有水平结构的g束的模空间上构造了局部系统作为hecke -本征束的本征系统。他们构造了约化群的Kloosterman轴，实现了几个例外代数群作为几何单群。该项目的目的是推广Heinloth， Ngô和Yun的构造，以获得新的刚性局部系统类别。此外，我们希望重新获得某些已知的例子。特别地，我们希望得到Katz的超几何轴的自同构解释。这些是上面提到的高斯超几何方程的l进类比和推广。