Families of trianguline representations, finite slope spaces and eigenvarieties

这个项目位于代数几何，代数数论和自同构形式理论之间的十字路口。我想研究p进伽罗瓦群的p进表示族，更准确地说，是所谓的三角表示类。本研究一方面提供了求解Kisins有限斜率空间的新方法，另一方面推广了这一构造。然后我想比较得到的空间与特征变，即与自同构形式的p进族参数化的空间。进一步，我将讨论它与朗兰兹纲领和方丹-马祖尔猜想的联系，即与伽罗瓦表示和自同构形式之间的推测联系。

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.