Mohamed Moakher (University of Pittsburgh)
Event Date
2025-04-09
Event Time
02:30 pm ~ 04:00 pm
Event Location
Wachman Hall 412
Abstract: 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.