Weight k Bianchi modular form?

Question

A Bianchi modular form, roughly speaking, is a modular form that is defined over SL$_2(K)$ where $K$ is an imaginary qudratic field.

Various authors such as in http://www.lmfdb.org/knowledge/show/mf.bianchi.bianchimodularforms provide details for weight 2 vector-valued Bianchi forms. Specfically, I'd like to know if there is a reference for

1. Weight $k$ scalar-valued Bianchi modular forms
2. General Fourier expansions for weight $k$ Bianchi modular forms
3. Base change lifts of modular forms from $\mathbb Q$ to $K$

Preferably in clasical rather than adelic language, as I am interested in direct calculations.

Answer 1

Here is a representation-theoretic explanation why the idea of weight $k$ scalar-valued Bianchi modular forms doesn't work.

We can think of classical modular forms of weight $k$ as functions on $\mathrm{SL}_2(\mathbb{R})$ that are left-invariant by $\mathrm{SL}_2(\mathbb{Z})$ and transform in a certain way on the right via the $k$-th irreducible representation of $\mathrm{SO}(2)$, which is just the character $$\begin{pmatrix} \cos \theta & \sin \theta \\\ -\sin \theta & \cos \theta \end{pmatrix} \mapsto e^{ik\theta}.$$ Since $\mathrm{SL}_2(\mathbb{R}) / \mathrm{SO}(2) \cong \mathbb{H}$, we can think of these as functions on $\mathbb{H}$.

Adèlically, you can think of this as an automorphic form on $\mathrm{GL}_2(\mathbb{A}_{\mathbb{Q}})$ that is left-invariant by $\mathrm{GL}_2(\mathbb{Q})$, right-invariant under $K_{\mathrm{fin}}$, the maximal compact subgroup of $\mathrm{GL}_2$ of the finite adèles (embedded in $\mathrm{GL}_2(\mathbb{A}_{\mathbb{Q}})$), and transforms under $K_{\infty} = \mathrm{O}(2)$ via the action of an irreducible representation. The reason why the corresponding classical form is scalar-valued is that $\mathrm{SO}(2)$ is abelian, so all irreducible representations are characters, which are one-dimensional representations.

(As an aside, representations of $\mathrm{O}(2)$ can be two-dimensional, but this should means we should really think of scalar-valued holomorphic cusp forms as being vector-valued pairs of holomorphic and antiholomorphic cusp forms, and one is completely determined by the other, so there's no harm in just working with holomorphic cusp forms.)

For Bianchi modular forms, on the other hand, we start adèlically with $\mathrm{GL}_2(\mathbb{A}_{\mathbb{Q}(i)})$ that is left-invariant by $\mathrm{GL}_2(\mathbb{Q}(i))$, right-invariant under $K_{\mathrm{fin}}$, the maximal compact subgroup of $\mathrm{GL}_2$ of the finite adèles (embedded in $\mathrm{GL}_2(\mathbb{A}_{\mathbb{Q}(i)})$), and transforms under $K_{\infty} = \mathrm{U}(2)$ via the action of an irreducible representation. Classically, we can think of this as a scalar-valued function on $\mathrm{SL}_2(\mathbb{C})$ that is left-invariant by $\mathrm{SL}_2(\mathbb{Z}[i])$ and transforms in a certain way on the right via an irreducible representation of $\mathrm{SU}(2)$.

The problem now is that if this irreducible representation $\mathrm{SU}(2)$ is not one-dimensional (which, unlike for $\mathrm{SO}(2)$, can actually happen), we cannot think of these as scalar-valued functions on $\mathrm{SL}_2(\mathbb{C}) / \mathrm{SU}(2) \cong \mathbb{H}^3$, because the action of $\mathrm{SU}(2)$ is not by a character. Rather, this scalar-valued automorphic form on $\mathrm{SL}_2(\mathbb{C})$ generates a finite-dimensional vector space (as a representation of $\mathrm{SU}(2)$), and we can choose a basis $f_1,\ldots,f_m$ of this vector space and construct a vector-valued automorphic form $(f_1,\ldots,f_m)$ on $\mathbb{H}^3$.