Computing Congruences of Modular Forms and Galois Representations Modulo Prime Powers

TL;DR

This paper develops algorithms for computing congruences of modular forms and Galois representations modulo prime powers, providing new computational tools and results that inform liftability and level raising phenomena.

Contribution

It introduces algorithms for determining common roots of polynomials modulo prime powers and applies them to study congruences of modular forms and Galois representations.

Findings

Algorithms for root commonality modulo prime powers

Computational results on liftability of modular forms

Insights into level raising and congruences

Abstract

This article starts a computational study of congruences of modular forms and modular Galois representations modulo prime powers. Algorithms are described that compute the maximum integer modulo which two monic coprime integral polynomials have a root in common in a sense that is defined. These techniques are applied to the study of congruences of modular forms and modular Galois representations modulo prime powers. Finally, some computational results with implications on the (non-)liftability of modular forms modulo prime powers and possible generalisations of level raising are presented.