Bianchi Modular Forms over Imaginary Quadratic Fields with arbitrary class group

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 ${\mathrm{GL}}_2{\mathcal{O}}_K$ 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carried out for fields of class number $1$, $2$, and $3$ by the first author, and by his students Bygott and Lingham in unpublished theses.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}).

31 pages