Lifting ethereal Bianchi modular forms

Abstract

Mod $p$ Hecke eigenforms for congruence subgroups of $\operatorname{SL}_2(\mathbb{Z})$ 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.