Quantitative level lowering for modular forms

Mohamed Moakher (University of Pittsburgh)

Given a Hilbert modular form f of weight two over a totally real field F, we can associate to it a finite module Phi(f) known as the congruence module for f, which measures the congruences that f satisfies with other forms. When f is transferred to a quaternionic modular form f_D over a quaternion algebra D via the Jacquet-Langlands correspondence, we can similarly define a congruence module Phi(f_D) for f_D. Pollack and Weston proposed a quantitative relationship between the sizes of Phi(f) and Phi(f_D), expressed in terms of invariants associated to f and D. In this talk, I will outline the ideas underlying the proof of this relationship. The approach combines a method of Ribet and Takahashi with new techniques introduced by Böckle, Khare, and Manning.