Computational aspects of modular forms and p-adic Galois representations

自守型理论是代数数论的核心问题之一。它有着悠久的历史，同时也是最繁荣的课题之一，它取得了诸如费马最后定理的证明、佐藤-泰特猜想或塞尔猜想等惊人的成就。目前正在探索许多新的方向。基本的计算机算法是一个宝贵的工具，可以产生进一步推进理论的实验数据。本建议提供了一种方法来获得和实现这样的算法。它还解释了我们希望进行的一些有趣的实验。该方法适用于数字以及功能字段。它与经典自守形式的关系可以用朗兰兹纲领来解释--而且常常是用图解的形式。其目的是通过它们的Hecke特征系统来计算与自守相关的伽罗瓦表示。其关键工具是利用余紧算术子群对Bruhat-Tits建筑群的幂指数进行算法计算。其结果是一个有限的单纯集，因此是一个有限的组合对象。自守形式是局部系统在这些有限元上的截面。他们的Hecke特征值系统可以组合计算。

The theory of automorphic forms is one of the core topics of algebraic number theory. It has a long history and is at the same time, with such spectacular successes as the proof of Fermat s last theorem, the Sato-Tate conjecture or the Serre conjecture, one of the most flourishing topics. Presently many new directions are explored. Basic computer algorithms axe an invaluable tool to yield experimental data which further advance the theory. The present proposal provides one approach to obtain and implement such algorithms. It also explains a number of interesting experiments which we hope to perform. The approach applies to number as well as function fields. Its relation to classical automorphic forms is explained by the Langlands program - and in often conjectural. The intention is to compute Galois representations associated to automorphic via their Hecke eigensystems. The key tool is the algorithmic computation of quotients of Bruhat-Tits buildings by cocompact arithmetic subgroups. The result is a finite simplicial set and thus a finite combinatorial object. Automorphic forms are sections of local systems on these finite quotients. Their systems of Hecke eigenvalues can be computed combinatorially.