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PreprintCC BY 4.0arXiv

Lifting ethereal Bianchi modular forms

  • Lewis Combes
  • Håvard Damm-Johnsen
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Abstract

Mod p Hecke eigenforms for congruence subgroups of SL2(ℤ) which do not lift to characteristic $0$ have received much attention, and were named ethereal modular forms by G. Schaeffer. In this paper, we study liftability for mod p Bianchi modular forms: computations suggest that a lift to characteristic zero may always be possible by increasing the level, as in the case of classical modular forms. We develop efficient algorithms for computing degeneracy maps in cohomology, which allow us to rigorously verify eigenvalue congruences. For small primes and certain fields, we also prove that lifts always exist as a consequence of the recent modularity theorem due to Caraiani-Newton. A Magma implementation, complemented by LMFDB data due to Cremona, shows that in favourable parameter ranges, most mod p Bianchi modular forms lift.

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Cite

BibTeXRIS
BibTeX
@misc{combes2026lifting,
  title = {{Lifting ethereal Bianchi modular forms}},
  author = {Lewis Combes and Håvard Damm-Johnsen},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.07445},
  eprint = {2610.07445},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.07445},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Lifting ethereal Bianchi modular forms
AU  - Lewis Combes
AU  - Håvard Damm-Johnsen
PY  - 2026
DA  - 2026/10/05
PB  - arXiv
DO  - 10.48550/arxiv.2610.07445
UR  - https://arxiv.org/abs/2610.07445
AB  - Mod p Hecke eigenforms for congruence subgroups of SL_2(ℤ) which do not lift to characteristic $0$ have received much attention, and were named ethereal modular forms by G. Schaeffer. In this paper, we study liftability for mod p Bianchi modular forms: computations suggest that a lift to characteristic zero may always be possible by increasing the level, as in the case of classical modular forms. We develop efficient algorithms for computing degeneracy maps in cohomology, which allow us to rigorously verify eigenvalue congruences. For small primes and certain fields, we also prove that lifts always exist as a consequence of the recent modularity theorem due to Caraiani-Newton. A Magma implementation, complemented by LMFDB data due to Cremona, shows that in favourable parameter ranges, most mod p Bianchi modular forms lift.
ER  -