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

最近在算术几何和各种主题的突破在朗兰兹纲领的精神建立和假定某些几何对象之间的深刻对应关系：模和自守形式和某些数论对象：伽罗瓦表示。几何方面往往是服从计算和明确的性质的对应关系也数论对象成为计算访问。这项建议的目的涉及调查这些几何和算术对象直接或通过对应以及扩展的对应新的情况下。一个特别强调的是放在这些方面的有限域和模素数权力。具体的问题处理的模块化以及水平和权重优化的伽罗瓦表示模素数权力和明确的决心，当地伽罗瓦表示重视古典和希尔伯特模块化形式在所有素数。将采用的方法是实验，算法和理论和进展，预计从这些的相互作用。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对其他研究人员有用。

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.