Level lowering and Shapiro's conjecture

Let E/Q be an elliptic curve over the rationals. One can associate two rational integers that measure the ramification of this elliptic curve over various primes, the conductor NE and the minimal discriminant ?E. The Szpiro's conjecture states that for any e>0 there exists a constant Ce>0 such that |?E| < Ce (NE)6+e. This conjecture is equivalent to the ABC-conjecture and, if true, would imply solutions to many Diophantine equations. A consequence of Szpiro's conjecture is that we can bound vp(?E) for the largest prime p dividing ?E. In this talk I will show how a generalization of Ken's level-lowering result can be used to bound vp(?E).