Computational aspects of modular forms and p-adic Galois representations

自同构形式理论是代数数论的核心问题之一。它有着悠久的历史，同时，凭借诸如费马大定理的证明、佐藤-塔特猜想或塞尔猜想等令人瞩目的成功，它是最繁荣的课题之一。目前正在探索许多新的方向。基本的计算机算法是产生进一步推进理论的实验数据的宝贵工具。本提案提供了一种获得和实现这种算法的方法。它还解释了我们希望进行的一些有趣的实验。这种方法既适用于数字字段，也适用于函数字段。它与经典自同构形式的关系由朗兰兹纲领来解释——而且常常是推测性的。目的是通过其Hecke特征系统计算与自同构相关的伽罗瓦表示。关键工具是用紧算术子群计算Bruhat-Tits建筑商的算法。结果是一个有限的简单集合，因此是一个有限的组合对象。自同构形式是这些有限商上的局部系统的部分。它们的赫克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.