Explicit Geometric Langlands Correspondence for Rigid Local Systems

刚性局部系统最著名的例子可能是高斯超几何方程在复投影线P^1上的解层，该线在0、1和无穷远处穿孔。在这种情况下，刚性条件说，这个解层（这是一个局部系统）是由穿刺周围的局部单值同构决定的。这种局部单值性在一点上被给出为微分方程的基本解与其沿沿着围绕缺失点的简单路径的解析延拓之间的线性关系。高斯方程具有所有奇点都是正则奇点的特殊性质。粗略地说，这意味着解服从某种增长条件。刚性局部系统所产生的定期奇异微分方程以这种方式已发现应用在逆伽罗瓦理论和建设动机与特殊motivic伽罗瓦群。更一般地，可以分别为微分方程定义类似的概念。平凡向量丛上不一定正则奇异的联络。在这种情况下，这样的联系被称为刚性的，如果它的同构类是唯一确定的局部形式数据。基本上有两种方法来构造刚性连接，在有限基域的情况下，l-adic局部系统。其中第一个是卡茨和阿林金的定理，该定理指出，P^1的开子集上的任何不可约刚性联络都可以通过迭代傅里叶-拉普拉斯变换、用秩为1的联络扭转或通过莫比乌斯变换进行坐标变换，从秩为1的联络构造出来。在正则单数（resp。tamely分歧）的情况下，可以用Katz定义的中间卷积代替Fourier-Laplace变换。Dettweiler和Reiter曾用这种方法对具有单值群（G_2型单例外代数群）的光滑分歧刚性局部系统进行了分类。此外，我还用这种方法构造了具有G_2型微分Galois群的新的刚性不规则联络。第二种建立严格的地方制度的方法如下。Heinloth，Ngô & Yun利用几何Langlands对应在具有水平结构的G-丛的模空间上构造局部系统作为Hecke-本征层的本征系统。他们构造了约化群的Kloosterman层，实现了几个特殊的代数群作为几何单值群。这个项目的目的是推广Heinloth，Ngô & Yun的构造，以获得新的刚性局部系统类。此外，我们希望重新获得某些已知的例子。特别是我们希望获得自守解释卡茨的超几何层。这些是上述高斯超几何方程的l-adic模拟和推广。

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.