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

• Antragsteller: Professor Dr. Gabor Wiese
• Fachliche Zuordnung: Mathematik
• Förderung: Förderung von 2010 bis 2014
• Projektkennung: Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 171737983

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.

• DFG-Verfahren: Schwerpunktprogramme
• Teilprojekt zu: SPP 1489: Algorithmic and Experimental Methods in Algebra, Geometry and Number Theory
• Internationaler Bezug: Luxemburg