AI 中文总结
本文研究模 p Bianchi 模形式的提升问题，开发高效算法验证特征值同余，并借助模性定理证明某些情形下提升存在，实验表明多数模形式可提升。
AI 中文摘要
对于 $\operatorname{SL}_2(\mathbb{Z})$ 的同余子群，模 $p$ Hecke 本征形式若不能提升到特征 $0$，则受到广泛关注，G. Schaeffer 将其命名为以太模形式。本文研究模 $p$ Bianchi 模形式的可提升性：计算表明，如同经典模形式的情形，通过增加水平，提升到特征零可能总是可行的。我们开发了计算上同调中退化映射的高效算法，从而能够严格验证特征值同余。对于小素数和某些域，我们还借助 Caraiani-Newton 最近的模性定理证明了提升总是存在。结合 Cremona 提供的 LMFDB 数据，Magma 实现表明，在有利的参数范围内，大多数模 $p$ Bianchi 模形式可以提升。
英文摘要
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.
Comments26 pages, 9 tables, comments welcome!
