What are some applications/consequences of level lowering of Galois representations? I understand the application of Ribet's theorem in the proof of Fermat's last theorem but I am wondering what other kinds of results can follow from knowing a given modular representation, of level $Np$, also arises as a modular representation of level $N$. Are there other well known results whose proof uses such a result?
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2$\begingroup$ Beyond Fermat, a few other equations have been solved with the same method, notably $A^n+B^n = 2^r C^n$ (in work of Ribet and Darmon-Merel). This is discussed in Henri Cohen's book (Number Theory, volume II: Analytic and Modern Tools), if I recall correctly. $\endgroup$– Vesselin DimitrovCommented Apr 7, 2016 at 5:02
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First, a small clarification: level-lowering tells you that a modular representation in level $Np$ occurs in level $N$ only if by occurs you mean "is congruent to modulo $p$".
That said, my answer to your question is essentially trivial: level-lowering is useful every time you have a property $(P)$ which is known to satisfy $$P(f)\textrm{ true}\Rightarrow P(g)\textrm{ true}$$ provided $f\equiv g\mod p$ and its usefulness is in reducing the general case to the minimally ramified case (that is to the case where $\rho_f$ is no more ramified than $\bar{\rho}_f$).
For a concrete example (for the property $P$ "The Iwasawa Main Conjecture is true") see (among many other) Variation of Iwasawa invariants in Hida families (Invent. math. 163) by M.Emerton, R.Pollack and T.Weston.
