Congruences between modular forms modulo prime powers

Abstract

Given a prime $p\ge 5$ and an abstract odd representation ${\rho }_n$ with coefficients modulo $p^n$ (for some $n\ge 1$) and big image, we prove the existence of a lift of ${\rho }_n$ to characteristic $0$ whenever local lifts exist (under minor technical conditions). Moreover, our results allow to chose the lift's inertial type at all primes but finitely many (where the lift is of Steinberg type).

We apply this result to the realm of modular forms, proving a level lowering theorem modulo prime powers and providing examples of level raising. An easy application of our main result proves that given a modular eigenform $f$ whose Galois representation is not induced from a character (i.e., $f$ has no inner twists), for all primes $p$ but finitely many, and for all positive integers $n$, there exists an eigenform $g\ne f$, which is congruent to $f$ modulo $p^n$.