Elliptic curves defined over the rational numbers arise from certain modular forms. This is the celebrated Modularity theorem of Wiles et al. Prior to this development, Ribet had proved a level lowering theorem, thanks to which one is able to optimize the level of the modular form in question. Ribet's theorem combined with the modularity theorem of Wiles together imply Fermat's Last theorem.

In joint work with Ravi Ramakrishna, we develop some new techniques to prove level lowering results for more general Galois representations. We prove level lowering theorems for 2-dimensional residually irreducible Galois representations over totally real fields. These are associated to Hilbert modular forms. In particular, we give an alternate proof of Ribet's theorem. We also have conditional results for higher dimensional Galois representations associated to more general automorphic forms.