Explicit Geometric Langlands Correspondence for Rigid Local Systems

也许刚性局部系统最著名的例子是复射影直线P^1上的Gauss超几何方程的解鞘，它在0，1和无穷远处穿孔。在这种情况下，刚性条件表示这个解束(它是一个局部系统)由穿孔周围的局部单行决定为同构。在某一点上的局部单调性被给出为微分方程的基本解与其沿失点附近的简单路径的解析延拓之间的线性关系。高斯方程具有奇点都是正则奇异的特殊性质。粗略地说，这意味着解决方案受到一些增长条件的制约。由正则奇异微分方程导出的刚性局部系统在伽罗瓦逆理论和具有特殊动机的伽罗瓦群的构造中得到了应用。更一般地，人们可以分别为微分方程定义类似的概念。平凡向量丛上的联络不一定是正则奇异的。在这种情况下，如果它的同构类由局部形式数据唯一确定，则这种连接称为刚性。构造刚性连接实质上有两种方法。在有限基域的背景下，得到了L-进局域系统。其中第一个是Katz&Arinkin的一个定理，它指出P^1的开子集上的任何不可约刚性联络可以通过迭代傅立叶-拉普拉斯变换、用一阶联络扭转或通过Möbius变换由一阶联络构造。在常规单数中(分别为驯服分支)的情况下，可以用Katz定义的中间卷积来代替傅里叶-拉普拉斯变换。这一方法被Dettweeller&Reiter用来分类具有单角群的柔和分枝局部刚体系统G_2型简单例外代数群。此外，我还用这种方法构造了与G_2型可微Galois群的新的刚性不规则连接。Heinloth，Ngô和Yun利用几何朗兰兹对应构造了具有水平结构的G-丛的模空间上的局部系统作为Hecke-特征层的特征系统。他们构造了约化群的Krousterman层，将几个特殊的代数群实现为几何单调群.这个项目的目的是推广Heinloth，Ngô&Yun的构造，得到新的刚性局部系统.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超几何束的自同构解释。这些是上面提到的高斯超几何方程的L-ADDIC的类比和推广。

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.