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Bianchi Modular Forms over Imaginary Quadratic Fields with arbitrary class group

This paper presents algorithms and computational techniques for determining the space of Bianchi modular forms over imaginary quadratic fields with arbitrary class groups, extending previous work to include detailed examples such as $\mathbb{Q}(\sqrt{-17})$ and providing extensive data now available in the LMFDB.

Original authors: John Cremona, Kalani Thalagoda, Dan Yasaki

Published 2026-02-03

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CC BY 4.0

Original authors: John Cremona, Kalani Thalagoda, Dan Yasaki

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to map a vast, invisible landscape. In the world of mathematics, this landscape is a special kind of 3D space called hyperbolic space. The paper you are asking about is a guidebook written by three mathematicians (John Cremona, Kalani Thalagoda, and Dan Yasaki) on how to navigate this space to find hidden treasures called Bianchi modular forms.

Here is a simple breakdown of what they did, using everyday analogies.

1. The Landscape and the Map

Think of the "hyperbolic space" as a giant, curved room. Inside this room, there are invisible walls and floors that form a complex 3D puzzle. Mathematicians call this a tessellation (like tiling a floor, but in 3D).

To study this space, the authors needed a way to break it down into manageable pieces. They used two different "construction kits" to build these 3D puzzles:

Kit A: Uses a method based on "perfect shapes" (perfect Hermitian forms).
Kit B: Uses an old algorithm from the 1970s (Swan's algorithm) that acts like a "pseudo-Euclidean algorithm" (a fancy way of saying it's a method for simplifying fractions, but applied to 3D shapes).

The authors built two separate computer programs using these different kits. The fact that they got the exact same results using two completely different construction methods proves their map is accurate.

2. The Treasure Hunt: Finding "Modular Forms"

The "treasures" they are looking for are Bianchi modular forms.

The Analogy: Imagine these forms as musical notes or radio signals that can only exist if the room (the space) has a very specific shape.
The Challenge: In the past, mathematicians could only easily find these signals in rooms that were very simple (like a room with no "twists" or "loops" in its structure). These simple rooms correspond to number fields with a "class number" of 1, 2, or 3.
The Breakthrough: This paper focuses on a much harder room: $K = \mathbb{Q}(\sqrt{-17})$ . This room has a "class group" of order 4. In our analogy, this means the room has a more complex twist or loop that makes it harder to navigate. The authors successfully mapped this complex room and found the signals inside it.

3. The "Hecke" Operators: The Sorting Machine

Once they found the signals (the modular forms), they needed to sort them. They used a machine called Hecke operators.

The Analogy: Think of the Hecke operators as a giant sorting machine or a set of filters. You feed a signal in, and the machine tells you its "frequency" or "signature."
The Problem: Usually, you need to look at the whole 3D room to get the full signature. However, the authors discovered a shortcut. They found that if you only look at the "main floor" of the room (the principal component), you can still figure out the full signature of the signal, provided you do a little bit of extra math to fill in the gaps. This is like being able to identify a whole song just by listening to the bass line.

4. The "Self-Twist" Mystery

One of the most interesting things they found is a phenomenon called self-twist.

The Analogy: Imagine a signal that, when you rotate the room by 180 degrees, sounds exactly the same, but with a slight "flip" in its phase.
The Discovery: In most cases, the signals are unique. But in this specific complex room ( $\mathbb{Q}(\sqrt{-17})$ ), some signals have this "self-twist" property. This makes them harder to find because half of their "notes" cancel out to zero. The authors developed a way to detect these tricky signals by checking if the room's structure allows for such a twist.

5. Connecting to Elliptic Curves (The "Real World" Link)

The ultimate goal of finding these abstract signals is to connect them to elliptic curves.

The Analogy: An elliptic curve is like a specific type of equation that describes a shape (often used in cryptography). The authors found a signal (a modular form) and proved that it is the "fingerprint" of a specific elliptic curve living in this number field.
The Proof: They didn't just guess; they used a rigorous mathematical test (the Serre-Faltings-Livné method) to prove that the signal and the curve are actually the same thing in disguise. They did this for a specific curve labeled 2.0.68.1-7.2-a.2.

6. The Big Picture

The authors didn't just stop at one room. They ran their code on 763 different rooms (imaginary quadratic fields) with varying levels of complexity.

They created a massive database (the LMFDB, or L-functions and Modular Forms Database) where anyone can look up the "dimensions" of these rooms and the "signatures" of the signals found inside.
They confirmed that their two different computer programs agree on the results, ensuring the data is trustworthy.

Summary

In short, this paper is a computational tour de force. The authors built two independent, high-tech "scanners" to map complex 3D mathematical spaces that were previously too difficult to navigate. They successfully found hidden patterns (modular forms) in a tricky environment, proved that these patterns match specific mathematical shapes (elliptic curves), and published a massive catalog of their findings for the world to use. They didn't just solve a puzzle; they built a better way to solve puzzles for everyone else.

Technical Summary: Bianchi Modular Forms over Imaginary Quadratic Fields with Arbitrary Class Group

Problem Statement
The paper addresses the computational challenge of determining the space of Bianchi modular forms of level $\Gamma_0(\mathfrak{n})$ over an arbitrary imaginary quadratic field $K = \mathbb{Q}(\sqrt{-d})$ . While previous work by the first author and his students successfully computed these spaces for fields with class numbers 1, 2, and 3, and for specific cases of higher class numbers, a general algorithm capable of handling fields with arbitrary class groups (specifically those with non-trivial 2-rank and even class numbers) was lacking. The primary difficulty lies in the fact that for fields with even class numbers, the space of modular forms decomposes into components associated with different ideal classes, and the action of the full Hecke algebra on the principal homology (which is computationally accessible) does not immediately yield the full eigensystems due to the existence of "self-twists" and non-principal operators.

Methodology
The authors develop and implement two independent algorithms to compute Bianchi modular forms as Hecke modules. Both methods rely on the isomorphism between the space of cuspidal Bianchi modular forms and the homology of the quotient of hyperbolic 3-space, $H_1(H_3/\Gamma_0(\mathfrak{n}), \mathbb{Q})$ .

Tessellation of Hyperbolic 3-Space:
- Method A (Swan's Algorithm): Utilizes a "pseudo-Euclidean algorithm" derived from Swan's work to generate a tessellation of $H_3$ by ideal polyhedra. This approach generalizes the continued fraction algorithm used for Euclidean fields.
- Method B (Perfect Hermitian Forms): Utilizes the reduction theory of perfect Hermitian forms (following Ash and Koecher) to construct a tessellation.
- Both methods require a precomputation stage to generate the tessellation data for a specific field $K$ , which is then used to compute the homology.
Hecke Action and Eigensystem Recovery:
- The algorithms compute the action of principal Hecke operators (those corresponding to principal ideals or squares of ideals) on the principal component of the homology, $H_1(X_0(\mathfrak{n}), \mathbb{Q})$ .
- A critical component of the methodology is the procedure to recover a full Hecke eigensystem (including non-principal operators and character information) from the principal homological eigensystem.
- Odd Class Number Case: If the class number $h_K$ is odd, every ideal class is a square, and the full eigensystem is determined uniquely (up to twist) by the principal eigenvalues.
- Even Class Number Case: When $h_K$ is even, the presence of unramified quadratic characters introduces ambiguity. The authors detail an algorithm to resolve signs for eigenvalues of non-square classes and to detect "self-twists" (where a form is invariant under twisting by a quadratic character). This involves analyzing the multiplicity of oldforms at higher levels and checking for vanishing eigenvalues in specific ideal classes.
- Atkin-Lehner Operators: The implementation includes code to compute Atkin-Lehner eigenvalues $W_{\mathfrak{q}}$ for exact divisors $\mathfrak{q} \parallel \mathfrak{n}$ , utilizing principal operators $T_{\mathfrak{a},\mathfrak{a}}W_{\mathfrak{q}}$ when $\mathfrak{q}$ is not principal.
Implementation:
- bianchi-progs: A C++ package implementing Method A (pseudo-Euclidean algorithm).
- Magma: A Magma implementation (by the third author, extended by the second) implementing Method B (perfect forms).
- The authors cross-verified results between these two implementations for consistency.

Key Contributions

Generalization: The paper extends the computation of Bianchi modular forms to imaginary quadratic fields with arbitrary class groups, moving beyond the restrictions of class numbers 1, 2, and 3.
Algorithmic Resolution of Self-Twists: The authors provide a rigorous method to handle forms admitting self-twists by unramified quadratic characters. They demonstrate how to identify these forms by analyzing the dimension of oldspaces and the vanishing of eigenvalues, a phenomenon that complicates the relationship between homological and full eigensystems.
Explicit Recovery of Eigensystems: The paper details the algorithmic steps to reconstruct the full Hecke eigensystem (including non-principal eigenvalues and character data) solely from the principal homological data, addressing the "sign ambiguity" in non-square ideal classes.
Proof of Modularity: The authors apply their computational framework to prove the modularity of a specific elliptic curve over $K = \mathbb{Q}(\sqrt{-17})$ (LMFDB label 2.0.68.1-7.2-a.2). They utilize the Serre-Faltings-Livné method (as implemented in [DGP10]) to verify that the Galois representations of the elliptic curve and the computed Bianchi newform are isomorphic.

Results

Field $K = \mathbb{Q}(\sqrt{-17})$ : The authors provide extensive data for this field, which has a cyclic class group of order 4.
- They computed dimensions and Hecke eigensystems for levels with norm up to 200 (and up to 1000 for rational newforms).
- They identified 1100 new Hecke eigensystems with unramified characters, of which 8 admit self-twists.
- They tabulated the decomposition of the newspace into irreducible subspaces under the Hecke algebra, distinguishing between trivial character ( $\chi_0$ ) and other characters ( $\chi_1, \chi_2, \chi_3$ ).
- They analyzed the structure of Hecke fields, showing cases where the full Hecke field is a quadratic extension of the principal Hecke field (e.g., $k_F = k_f(\sqrt{r})$ ).
Broader Scope: Computations have been performed for 763 imaginary quadratic fields with absolute discriminant up to 2500. Data for rational newforms (those with Hecke field $\mathbb{Q}$ ) is available in the L-functions and Modular Forms Database (LMFDB).
Consistency: The two independent implementations (C++ and Magma) produced consistent results for homology dimensions and Hecke eigenvalues for the tested ranges.

Significance
The paper claims significance primarily as a computational extension of existing theory. It provides the necessary algorithms and implementations to study Bianchi modular forms over fields where the class group structure prevents the simplifications used in previous work (e.g., fields with even class numbers). By successfully computing these spaces and proving the modularity of an elliptic curve over $\mathbb{Q}(\sqrt{-17})$ , the authors demonstrate that the computational framework is robust enough to handle the complexities of arbitrary class groups. The data generated fills a gap in the LMFDB for fields with higher class numbers and provides a foundation for further arithmetic investigations, such as the study of base changes and the modularity of elliptic curves over these fields. The work complements recent theoretical advances (such as those by Newton and Caraiani) by providing explicit computational evidence and data for cases not covered by general theorems.