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.

  • $\begingroup$ Cremona and his students often have quite elementary and explicit descriptions of these objects, e.g. see warwick.ac.uk/fac/sci/maths/research/interests/number_theory/…. It also depends on what "direct calculations" you want to do. $\endgroup$ – The Piper Jun 28 at 16:09
  • $\begingroup$ Cremona and his students typically work with modular symbols, and don't give any information about say, properties (1)–(3) that I'm looking for. $\endgroup$ – TA Wong Jun 28 at 16:22
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    $\begingroup$ The link I gave has an explicit formula (page $12$) for the Fourier expansion of weight $2$ forms. How are you possibly suggesting to compute anything without doing computations involving cohomology? $\endgroup$ – The Piper Jun 28 at 17:29
  • $\begingroup$ Actually, reading your question a little more closely and clicking on your link, I think you are laboring under some more serious misapprehensions. I'm not sure what you think a "scalar" Bianchi form is, but, in the context of that link, no such objects exist (well, except for constant functions). Perhaps you should say what you are actually interested in doing and we can work backwards from there. $\endgroup$ – The Piper Jun 28 at 20:07
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    $\begingroup$ See section 1 arxiv.org/pdf/1404.2100.pdf $\endgroup$ – TA Wong Jun 28 at 20:54

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 $\mathbb{H}^3$.

  • $\begingroup$ If $f,g$ are two modular forms of any weight and level then the $SL_2(\Bbb{Z})$-orbit of $f+g$ is a finite dimensional vector space and I think something comparable happens $\endgroup$ – reuns Jul 2 at 21:24
  • $\begingroup$ thanks! i just wasn't completely sure that there aren't any nontrivial one-dim reps of SU(2), but this indeed seems to be the case. $\endgroup$ – TA Wong Jul 3 at 14:54

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