Topics on modular Galois representations modulo prime powers

Abstract

This article surveys modularity, level raising and level lowering questions for two-dimensional representations modulo prime powers of the absolute Galois group of the rational numbers. It contributes some new results and describes algorithms and a database of modular forms orbits and higher congruences.

20 pages; this article supersedes arXiv:1203.6541 by the first author; accepted for publication in: Böckle, Decker, Malle (Editors). Algorithmic and Experimental Methods in Algebra, Geometry, and Number Theory