Topics on modular Galois representations modulo prime powers

TL;DR

This paper surveys key topics in modular Galois representations modulo prime powers, introduces new results, and provides algorithms and a database related to modular forms and higher congruences.

Contribution

It offers new results on modularity, level raising/lowering, and presents algorithms and a database for modular forms and higher congruences.

Findings

New results on modularity and level phenomena

Algorithms for computing modular forms orbits

A database of modular forms and higher congruences

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.