Bianchi Modular Forms over Imaginary Quadratic Fields with arbitrary class group

TL;DR

This paper develops algorithms to compute Bianchi modular forms over imaginary quadratic fields with arbitrary class groups, extending previous work limited to fields with small class numbers, and provides explicit examples including modularity of an elliptic curve.

Contribution

The paper introduces new computational techniques for Bianchi modular forms over fields with arbitrary class group, expanding the scope beyond class number 1, 2, and 3 cases.

Findings

Algorithms successfully compute Bianchi modular forms for complex class groups.

Explicit examples include modularity proof for an elliptic curve over 9-17.

Results are included in the LMFDB database.

Abstract

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 $\Gamma_0({\mathfrak{n}})$ 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 $\Gamma_0({\mathfrak{n}})$ 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carried out for fields of class number $1$, $2$, and $3$ by the first author, and by his students Bygott and Lingham in unpublished theses.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}).