An Exotic Congruence for Irregular Weight Picard Modular Forms

We construct a new type of congruence from irregular weight mod $p$ Picard modular forms (for $p$ inert) to ones of regular weight, analogous to the one obtained by Kansal–Levin–Savitt in the Hilbert setting. The congruence is obtained by using a geometric Jacquet–Langlands phenomenon, as first established by Vollaard, to transfer irregular weight mod $p$ Picard modular forms from the Picard modular surface to a Shimura set. We call this an exotic congruence, since it relates mod $p$ automorphic forms associated to different groups. This congruence will be used in upcoming work of the author proving mod $p$ local-global compatibility for irregular weight Picard modular forms (in the $p$ inert case).