Explicit Serre weights for modular Galois representations of U(2)
Abstract
Let p be an odd prime. Let F be a totally imaginary CM field and F+ be the maximal totally real subfield such that F/F+ is unramified at all finite places, and every place $v | p$ of F+ is split in F. We allow arbitrary ramification of p in F+. Let G be a rank two definite unitary group over F+ as defined in [2]. Fix a place p | p in F. In this article, we determine explicit local Serre weights at p attached to a modular mod p Galois representation ρ̅h of the absolute Galois group of F, associated to an eigenform h of G. The eigenform h has arbitrary weights and small slope at p. We also show analogous results for certain Hilbert modular eigenforms with arbitrary ramification of p in the totally real field. The eigenforms come from totally definite and indefinite quaternion algebras over totally real fields, split at places above p. In proving our results, we identify the non-singular submodules in quotients of certain symmetric power 𝔽q[M2(𝔽q)]-modules by Dickson polynomials and their twisted versions.
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BibTeX
@misc{ganguli2026explicit,
title = {{Explicit Serre weights for modular Galois representations of $U(2)$}},
author = {Abhik Ganguli and Suneel Kumar},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.37261},
eprint = {2610.37261},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.37261},
}
RIS
TY - UNPB
TI - Explicit Serre weights for modular Galois representations of U(2)
AU - Abhik Ganguli
AU - Suneel Kumar
PY - 2026
DA - 2026/10/27
PB - arXiv
DO - 10.48550/arxiv.2610.37261
UR - https://arxiv.org/abs/2610.37261
AB - Let p be an odd prime. Let F be a totally imaginary CM field and F^+ be the maximal totally real subfield such that F/F^+ is unramified at all finite places, and every place $v | p$ of F^+ is split in F. We allow arbitrary ramification of p in F^+. Let G be a rank two definite unitary group over F^+ as defined in [2]. Fix a place p | p in F. In this article, we determine explicit local Serre weights at p attached to a modular mod p Galois representation ρ̅_h of the absolute Galois group of F, associated to an eigenform h of G. The eigenform h has arbitrary weights and small slope at p. We also show analogous results for certain Hilbert modular eigenforms with arbitrary ramification of p in the totally real field. The eigenforms come from totally definite and indefinite quaternion algebras over totally real fields, split at places above p. In proving our results, we identify the non-singular submodules in quotients of certain symmetric power 𝔽_q[M_2(𝔽_q)]-modules by Dickson polynomials and their twisted versions.
ER -