Computational aspects of modular forms and p-adic Galois representations

SCHEME: INTER

CALL: 2013

DOMAIN: IS - Information and Communication Technologies

FIRST NAME: Gabor

LAST NAME: Wiese

INDUSTRY PARTNERSHIP / PPP: No

INDUSTRY / PPP PARTNER:

HOST INSTITUTION: University of Luxembourg

KEYWORDS:

START: 2013-08-01

END: 2016-07-31

WEBSITE: https://www.uni.lu

Submitted Abstract

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 project concern the investigation of these geometric and arithmetic objects either directly or through the correspondence. One main research line will lead to the computation of integral properties of p-adic modular Galois representations. Another research line will study the finiteness and describe the growth of sets of p-adic modular Galois representations modulo prime powers. Both lines are naturally related to level and weight optimisation modulo prime powers. They are also intimately linked to the p-adic coefficient fields of newforms about which numerous conjectures exist. These coefficient fields are strongly affected by the local coefficient field at p and a so called L-invariant. The latter will be explored by generating new data. Finally in weight two the integral local representation at good primes away from p will be studied by methods involving abelian varieties and their endomorphism rings.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 and a publicly available database will be computed. These new computer tools will be of service to other researchers as well

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