Quantitative level lowering for weight two Hilbert modular forms

Given a Hilbert modular form f over a totally real field F, we can associate to it a finite module P(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 P(f_D) for f_D. Pollack and Weston proposed a quantitative relationship between the sizes of P(f) and P(f_D), expressed in terms of invariants associated to f and D.

In this talk, I will present the proof of this relationship under some hypotheses. The proof combines a generalization of a result of Ribet and Takahashi with new techniques introduced by Böckle, Khare, and Manning.