For example, such a result was used
by Ribet for GL_2 (1990) to prove that
the Taniyama-Weil Conjecture implies Fermat's theorem.
In his thesis (1989), Taylor introduced
a general method to raise the level for
GL_2, to construct the global Langlands correspondance for
classical Hilbert modular forms. Bellaiche (2002)
generalized this for any rank one group, and used it to prove parts of the
Bloch-Kato conjecture. In a recent joint work, we obtained a general
statement for U(3).