Bianchi Modular Forms over Imaginary Quadratic Fields with arbitrary class group

Let $K$ be an imaginary quadratic field and let $\mathcal{O}_K$ be its ring of integers. For an integral ideal $\mathfrak{n}$ of $\mathcal{O}_K$, let $Γ_0({\mathfrak{n}})$ be the congruence subgroup of level ${\mathfrak{n}}$ consisting of matrices in $\operatorname{GL}_2{\mathcal{O}_K}$ that are upper triangular mod ${\mathfrak{n}}$. In this paper, we discuss techniques to compute the space of Bianchi modular forms of level $Γ_0({\mathfrak{n}})$ as a Hecke module in the case where $K$ has arbitrary class group. Our algorithms and computations extend and complement those carried out for fields of class number $1$, $2$, and $3$ by the first author, and by his students Bygott and Lingham in unpublished theses. We give details and several examples for $K=\mathbb{Q}(\sqrt{-17})$, whose class group is cyclic of order $4$, including a proof of modularity of an elliptic curve over this field. We also give an overview of the results obtained for a wide range of imaginary quadratic fields, which are tabulated in the L-functions and modular forms database (\href{https://www.lmfdb.org/}{LMFDB}).

💡 Research Summary

The paper develops and implements algorithms for computing spaces of Bianchi modular forms over arbitrary imaginary quadratic fields K, regardless of the structure of the class group Cl K. For an integral ideal 𝔫⊂𝒪_K, the authors consider the congruence subgroup Γ₀(𝔫)⊂GL₂(𝒪_K) and aim to determine the Hecke module S(𝔫) of cuspidal weight‑2 Bianchi forms with trivial (or unramified) character.

Two independent computational approaches are presented. The first follows Swan’s 1971 algorithm to obtain a tessellation of hyperbolic 3‑space ℍ³ by ideal polyhedra and uses a “pseudo‑Euclidean” reduction process (originally introduced by Whitley) to compute Hecke operators. The second builds on Ash’s theory of perfect Hermitian forms and Koecher’s reduction, employing Gunnells’s reduction theory to obtain a different tessellation and to compute Hecke operators via a more geometric reduction. Both methods require a one‑time pre‑computation of a tessellation for each field K; thereafter the rational homology H₁(ℍ³/Γ₀(𝔫),ℚ) is computed, and the full Hecke algebra T acts on it.

Hecke operators come in two families: the standard operators T_𝔞 for any integral ideal 𝔞, and the “double” operators T_{𝔞,𝔞} defined when 𝔞 is coprime to the level and 𝔞² is principal. These operators are graded by the class group: an operator of class c maps the homology component attached to class c′ into the component of class c′c⁻¹. Consequently, only the principal subalgebra T₁ (generated by operators attached to principal ideals) acts on the principal homology H₁(X₀(𝔫),ℚ). The authors give explicit formulas for all operators, extending earlier work of Cremona, Bygott, Lingham, Aranes and Whitley.

A key subtlety is the presence of self‑twists. When Cl K has even order, a Bianchi newform may admit a non‑trivial unramified quadratic character ψ such that the form is invariant under twisting by ψ. In that case, Hecke operators T_𝔞 with ψ(𝔞)=−1 act as zero, which complicates the linear algebra. The paper explains how to detect and handle such situations, proving that a newform can have at most one self‑twist because otherwise the attached ℓ‑adic Galois representation would have finite image, contradicting known results.

Implementation details are given for two software packages. The C++ library “bianchi‑progs” (available on GitHub) implements the Swan‑based method; the Magma package “BCP97” (developed by the third author and extended by the second) implements the Ash‑Koecher method. Both have been cross‑validated extensively: (i) for all fields with |disc K|≤2100 the homology dimensions at level (1) agree; (ii) for fields with |disc K|≤100 and norms N(𝔫)≤100 the homology dimensions match; (iii) for K=ℚ(√−17) (class group cyclic of order 4) the dimensions, cusp‑form spaces, and Hecke eigenvalues for N(𝔫)≤200 coincide in both implementations.

The paper devotes a detailed case study to K=ℚ(√−17). Because the class group is C₄, the adelic picture involves four copies of ℍ³, each acted on by a twisted congruence subgroup Γ(c)₀(𝔫). The authors compute dimension tables for levels with norm up to 1000 (using the C++ code) and verify them up to norm 200 with Magma. They exhibit explicit newforms, and for one of them they prove modularity of the elliptic curve with LMFDB label 2.0.68.1.7.2‑a.2. This extends earlier modularity results of Dieulefait‑Guerberoff‑Pacetti (which covered fields of class number 3) to a field of class number 4. The authors note that the general modularity theorem of Newton‑Caraiani (2025) does not apply here because X₀(15) over ℚ(√−17) has Mordell–Weil rank 2, whereas the theorem requires rank 0.

Finally, the authors summarize the extensive data now available in the L‑functions and Modular Forms Database (LMFDB). For every imaginary quadratic field with discriminant up to 2100, they have computed homology dimensions, cusp‑form dimensions, and Hecke eigenvalues for levels with norm ≤200 (and higher norms for selected fields). The data include spaces with trivial and all unramified characters, and tables of newforms with rational Hecke eigenvalues. The paper concludes with a discussion of future directions: handling higher‑weight Bianchi forms, exploring the interaction of multiple self‑twists, relating Bianchi newforms to higher‑dimensional abelian varieties, and extending the computational framework to more general number fields.

In summary, this work provides a robust, field‑independent algorithmic framework for computing Bianchi modular forms, validates it through two independent implementations, and enriches the LMFDB with a substantial new dataset, thereby opening new avenues for research in automorphic forms, Galois representations, and arithmetic geometry.