Families of trianguline representations, finite slope spaces and eigenvarieties

该项目位于代数几何、代数论和自同构形理论之间的十字路口。我想研究p-进Galois群的p-进表示的族，更准确地说，是所谓的三角表示类。这项研究一方面应该为Kisins有限斜率空间提供一种新的方法，另一方面也应该推广这种构造。然后，我想将所得到的空间与特征簇进行比较，即与参数化自同构形式的p-进数族的空间进行比较。进一步，我将讨论与朗兰兹程序和Fontaine-Mazur猜想的联系，即与Galois表示和自同构形之间的猜想联系。

This project lies on the crossroads between Algebraic Geometry, Algebraic Number theory and the theory of automorphic forms. I want to study families of p-adic representations of p-adic Galois groups, and, more precisely, the class of so called trianguline representations. This study should yield on the one hand a new approach to Kisins finite slope space, on the other hand it should generalize this construction. Then I want to compare the resulting spaces with eigenvarieties, i.e with spaces parametrizing p-adic families of automorphic forms. Further I will discuss the connection with the Langlands program and the Fontaine-Mazur conjecture, i.e. with the conjectural connection between Galois representations and automorphic forms.