Counting level-raising congruences using modular representation theory
Abstract
We introduce a new method for studying mod-ℓ congruences between eigenforms through the modular representation theory of PGL2(𝔽p). When p≡ ± 1 pmodℓ, we use this theory to construct and describe extra structures on spaces of modular forms with Γ0(p2)-level at p and a fixed mod-ℓ Galois representation. The structural results we obtain can be viewed as a refinement of classical level-raising theorems since they not only allow us to prove the existence of congruences, but also to count the number of such congruences. Our methods work equally well in the residually irreducible and residually reducible cases, allowing us to prove several new instances of congruences between Eisenstein series and cuspforms (as well as independently rederiving classical results of Mazur and more recent results of Lang--Wake). Notably, our approach proves these results without computing constant terms of Eisenstein series, without Galois deformation theory and R=𝕋 theorems, and without the Jacquet--Langlands correspondence.
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BibTeX
@misc{lang2026counting,
title = {{Counting level-raising congruences using modular representation theory}},
author = {Jaclyn Lang and Robert Pollack and Preston Wake},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.05144},
eprint = {2609.05144},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.05144},
}
RIS
TY - UNPB
TI - Counting level-raising congruences using modular representation theory
AU - Jaclyn Lang
AU - Robert Pollack
AU - Preston Wake
PY - 2026
DA - 2026/09/04
PB - arXiv
DO - 10.48550/arxiv.2609.05144
UR - https://arxiv.org/abs/2609.05144
AB - We introduce a new method for studying mod-ℓ congruences between eigenforms through the modular representation theory of PGL_2(𝔽_p). When p≡ ± 1 pmodℓ, we use this theory to construct and describe extra structures on spaces of modular forms with Γ_0(p^2)-level at p and a fixed mod-ℓ Galois representation. The structural results we obtain can be viewed as a refinement of classical level-raising theorems since they not only allow us to prove the existence of congruences, but also to count the number of such congruences. Our methods work equally well in the residually irreducible and residually reducible cases, allowing us to prove several new instances of congruences between Eisenstein series and cuspforms (as well as independently rederiving classical results of Mazur and more recent results of Lang--Wake). Notably, our approach proves these results without computing constant terms of Eisenstein series, without Galois deformation theory and R=𝕋 theorems, and without the Jacquet--Langlands correspondence.
ER -