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

在朗兰兹计划的精神下，算术几何和各种专题猜想的最新突破建立和假定了某些几何对象之间的深度对应：模形式和自同构形式与某些数论对象：伽罗瓦表示。几何方面通常服从于计算，并且由于对应关系的显性性质，数论对象也变得可以通过计算访问。这项建议的目标是直接或通过函件调查这些几何和算术对象，并将函件扩大到新案件。在有限域和模素数幂上特别强调了这些方面。具体问题涉及模素数幂的Galois表示的模性以及水平和重量的优化，以及在所有素数上附加于经典和Hilbert模形式的局部Galois表示的显式确定。将采用的方法包括实验方法、算法方法和理论方法，这些方法的相互作用有望取得进展。对于实验研究，算法将被开发并在计算机代数系统中实现。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.