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

有限域和模素数幂上的模伽罗瓦表示的算法和实验方面

• 批准号: 171737983
• 负责人: Professor Dr. Gabor Wiese
• 金额: --
• 依托单位: Faculty of Science, Technology and Medicine (FSTM)
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2010
• 资助国家: 德国
• 起止时间: 2009-12-31 至 2013-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/171737983?language=en
• 关键词: Algorithmic; experimental; aspects; modular; Galois

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.

算术几何的最新突破和本着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。几何方面通常易于计算，并且通过对应关系的明确性质，数论对象也变得可计算。该提案的目标涉及直接或通过对应关系以及将对应关系扩展到新案例来研究这些几何和算术对象。特别强调有限域和模素数幂的这些方面。具体问题涉及模素数幂的伽罗瓦表示的模块化以及级别和权重优化，以及所有素数处附加到经典和希尔伯特模形式的局部伽罗瓦表示的显式确定。所采用的方法包括实验方法、算法方法和理论方法，预计这些方法的相互作用将取得进展。对于实验研究，将在计算机代数系统中开发和实现算法。这些新的计算机工具也将为其他研究人员提供服务。