On congruences mod pm between eigenforms and their attached Galois representations

Given a prime p and cusp forms f1 and f2 on some Γ1(N) that are eigenforms outside Np and have coefficients in the ring of integers of some number field K, we consider the problem of deciding whether f1 and f2 have the same eigenvalues mod pm (where p is a fixed prime of K over p) for Hecke operators Tℓ at all primes ℓ∤Np. When the weights of the forms are equal the problem is easily solved via an easy generalization of a theorem of Sturm. Thus, the main challenge in the analysis is the case where the forms have different weights. Here, we prove a number of necessary and sufficient conditions for the existence of congruences mod pm in the above sense. The prime motivation for this study is the connection to modular mod pm Galois representations, and we also explain this connection.