Algorithmic and experimental aspects of modular Galois representations over finite fields and modulo prime powers

最近在算术几何方面的突破和基于朗兰兹纲领精神的各种主题猜想建立并假设了某些几何对象（模和自同构形式）与某些数论对象（伽罗瓦表示）之间的深层对应关系。几何方面通常可以进行计算，并且由于对应的显式性质，数论对象也可以通过计算获得。本提案的目标是直接或通过对应关系对这些几何和算术对象进行调查，并将对应关系扩展到新的情况。特别强调在有限域和模素幂上的这些方面。具体的问题涉及到伽罗瓦表示的模块性以及模素数幂的水平和权重优化，以及在所有素数上附加于经典和希尔伯特模形式的局部伽罗瓦表示的明确确定。将采用实验、算法和理论方法，并期望从这些方法的相互作用中取得进展。对于实验研究，算法将在计算机代数系统中开发和实现。这些新的计算机工具也将为其他研究人员服务。

Recent breakthroughs in Arithmetic Geometry and various topical conjectures in the spirit of the Langlands programme establish and postulate deep correspondences between certain geometric objects: modular and automorphic forms and certain number theoretic objects: Galois representations. The geometric side is often amenable to calculations and by the explicit nature of the correspondences also number theoretic objects become computationally accessible. The objectives of this proposal concern the investigation of these geometric and arithmetic objects either directly or through the correspondence as well as an extension of the correspondence to new cases. A special emphasis is placed on these aspects over finite fields and modulo prime powers. Concrete questions deal with the modularity as well as level and weight optimisation of Galois representations modulo prime powers and the explicit determination of the local Galois representations attached to classical and Hilbert modular forms at all primes. The methods to be employed are experimental, algorithmic and theoretical and progress is expected from the interplay of these. For the experimental study, algorithms will be developed and implemented in computer algebra systems. These new computer tools will be of service to other researchers as well.