Quantitative level lowering

We give a quantitative version of Ribet's famous level lowering result for modular forms. Specifically, we measure how certain congruence ideals change as we vary the level. By studying the deformation theory of the Galois representation attached to the modular form, we can use the numerical criterion of Wiles to relate congruence ideals to Selmer groups, and thereby reduce the problem to a calculation in Galois cohomology.