PreprintCC BY 4.0arXiv
An Exotic Congruence for Irregular Weight Picard Modular Forms
Abstract
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).
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BibTeX
@misc{sonne2026exotic,
title = {{An Exotic Congruence for Irregular Weight Picard Modular Forms}},
author = {Calle Sönne},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.26807},
eprint = {2610.26807},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.26807},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - An Exotic Congruence for Irregular Weight Picard Modular Forms
AU - Calle Sönne
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.26807
UR - https://arxiv.org/abs/2610.26807
AB - 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).
ER -