I often use the following theorem (that one can state more generally) in my research.
Let E/Q be an elliptic curve of conductor N corresponding to a modular form f(E), l a prime of good or multiplicative reduction for E, and \rho(l) the 2 dimensional mod l Galois representation given by the action on the l-torsion points. Suppose that the torsion subscheme E[l] extends to a finite flat group scheme over Z_l, and let p be a prime of multiplicative reduction for E such that \rho(l) is unramified at p (e.g. the number field (Q(E[l]) generated by the coordinates of the l-torsion points is unramified at p). Then there exists a modular form f of conductor N/p such that f is congruent to f(E) mod l (when f has Fourier coefficients over Z then this means that all but finitely many of the coefficents are congruent mod l); one can `lower the level' from N to N/p.
Does such a result hold for powers of primes? E.g. if this holds for the mod l^n representation (instead of the mod l) does one get a congruence mod l^n?
Best Answer
There's some slides from a talk by Ian Kiming here which discuss this question. He states a theorem (on slide number 8) corresponding to the existence of the map from a Hecke algebra at level N/(p^u) (where p^u is the largest power of p dividing N) to Z/ell^n Z. As buzzard says, it's not clear that this map will lift, but Kiming speculates that if you allow the weight of your modular form to vary you can find a char 0 lift.