Bianchi Modular Forms over Class Number 4 Fields
Kalani Thalagoda, Dan Yasaki
TL;DR
For $F = \mathbb{Q}(\sqrt{-17})$—a field with class number $4$—the paper develops and implements an algorithm to compute Bianchi modular forms as Hecke eigensystems, extending prior class-number-$1$ and low-class-number work. It leverages the homology of hyperbolic $3$-space quotients via Ash–Koecher tessellations and computes principal Hecke operators using $(\mathfrak{a},\mathfrak{b})$-matrices to extract eigenforms and their Hecke data, including twists by the class group. The results comprise extensive newform tables, analyses of base-change phenomena, and a modularity example of a rational elliptic curve over $F$, marking the first such computations for a class-number-$4$ imaginary quadratic field and providing a robust dataset for the LMFDB. The work establishes a data-driven framework for extending Bianchi modular form computations to fields with nontrivial class groups and larger discriminants, enabling deeper exploration of connections to Hilbert modular forms and elliptic curves over imaginary quadratic fields.
Abstract
Let $F$ be an imaginary quadratic field, and let $\mathcal{O}_F$ be its ring of integers. For any ideal $\mathfrak{n} \subset \mathcal{O}_F$, let $Γ_0(\mathfrak{n})$ be the congruence subgroup of level $\mathfrak{n}$ consisting of matrices that are upper triangular mod $\mathfrak{n}$. In this paper, we develop techniques to compute spaces of Bianchi modular forms of level $Γ_0(\mathfrak{n})$ as a Hecke module in the case where $F$ has cyclic class group of order $4$. This represents the first attempt at such computations and complements work for smaller class numbers done by Cremona and his students Bygott, Lingham \cite{bygott,lingham}. We implement the algorithms for $F = \mathbb{Q}(\sqrt{-17})$. In our results we observe a variety of phenomena.