Bianchi modular forms (reviewed)

Given $\gamma=\begin{pmatrix} a & b \\ c&d \end{pmatrix} \in \textrm{PSL}_2(\mathbb{C})$ and $z=(x,y) \in \mathcal{H}_3$, hyperbolic 3-space, let us introduce the multiplier system $$J(\gamma, z):= \begin{pmatrix} cx+d & -cy \\ \bar{c}y & \overline{cx+d}\end{pmatrix}.$$

Given a function $F: \mathcal{H}_3 \rightarrow \C^{k+1}$ and $\gamma \in \textrm{PSL}_2(\mathbb{C})$, we define the "slash operator" $$(F |_k\gamma)(z):=Sym^k(J(\gamma, z)^{-1}) \ F(\gamma z),$$ where $Sym^k$ is the symmetric $k^{th}$ power of the standard representation of $\textrm{PSL}_2(\mathbb{C})$ on $\mathbb{C}^2$.

The center of the universal enveloping algebra of the Lie algebra associated to the real Lie group $\textrm{PSL}_2(\mathbb{C})$ is generated by two elements (Casimir operators) $\Psi, \Psi'$. These act on real analytic functions $F: \mathcal{H}_3 \rightarrow \mathbb{C}^{k+1}$ as differential operators.

Let $K$ an imaginary quadratic field and $\mathcal{O}_K$ be its ring of integers. Let $\Gamma$ be a congruence subgroup of a Bianchi group $\textrm{PSL}_2(\mathcal{O}_K)$. A Bianchi modular form for $\Gamma$ with weight $k$ is a real analytic function $F: \mathcal{H}_3 \rightarrow \mathbb{C}^{k+1}$ with the properties

1. $F|_k\gamma = F$ for every $\gamma \in \Gamma$,

2. $\Psi F = 0$ and $\Psi' F = 0$,

3. $F$ has at worst polynomial growth at each cusp of $\Gamma$.

As there is no complex structure on $\mathcal{H}_3$, the notion of "holomorphicity" is not available. Property (2) should be considered as "harmonicity".

The set $M(\Gamma,k)$ of Bianchi modular forms for $\Gamma$ with weight $k$ is a finite dimensional complex vector space.

Fourier-Bessel expansions

As the hyperbolic 3-fold $\Gamma \backslash \mathcal{H}_3$ is non-compact, $\Gamma$ contains parabolic elements. It follows from property (1) that $F$ is a periodic function in the $x=(x_1,x_2)$-variable. It follows that the $F$ has a Fourier-Bessel expansion of the form $$F(x,y)=\sum_{0 \not =\alpha \in \mathcal{O}_K}c(\alpha) y^2 \mathbb{K}\left ( \dfrac{4\pi|\alpha|y}{\sqrt{|\triangle|}} \right ) \psi\left (\dfrac{\alpha x}{\sqrt{\triangle}} \right )$$ where $$\psi(x)=e^{2\pi(x+\bar{x})}$$ and
$$\mathbb{K}(t)=\left ( -\dfrac{i}{2}K_1(y),K_0(y),\dfrac{i}{2}K_1(y) \right)$$ with $K_0,K_1$ are the hyperbolic Bessel functions satisfying the differential equation $$\dfrac{dK_j}{dy^2}+\dfrac{1}{y}\dfrac{dK_j}{dy}-\left ( 1+\dfrac{1}{y^{2j}}\right )K_j = 0, \ \ \ \ j=0,1$$ and decreases rapidly at infinity.

Weight 2 Bianchi modular forms

The weight 2 Bianchi modular forms are particularly important in regard to their conjectural connections with abelian varieties of $\textrm{GL}_2$-type. In the weight 2 case, we have $F: \mathcal{H}_3 \rightarrow \mathbb{C}^3$ and $$(F |_k\gamma)(z)=\dfrac{1}{|r|^2+|s|^2} \begin{pmatrix} \bar{r}^2 & 2\bar{r}s & s^2 \\ -\bar{r}\bar{s} & |r|^2-|s|^2 & rs \\ \bar{s}^2 & -2r\bar{s} & r^2 \end{pmatrix} F(\gamma z)$$ where $\gamma=\begin{pmatrix} a &b \\ c&d \end{pmatrix}$ and $r=cx+d$ and $s=cy$.

Let $\beta_1:=-\frac{dx}{y}, \beta_2:= \frac{dy}{y}, \beta_3:=\frac{d\bar{x}}{y} $ be a basis of differential 1-forms on $\mathcal{H}_3$. A differential form $\omega$ is harmonic if $\Delta \omega =0$ where $\Delta=d \circ \delta + \delta \circ d$ is the usual Laplacian with $d$ being the exterior derivative and $\delta$ the codifferential operator. Then $\textrm{PSL}_2(\mathbb{C})$ acts on the space of differential 1-forms as $$\gamma \cdot {}^t(\beta_1,\beta_2,\beta_3)_{(z)} = Sym^2(J(\gamma,z)){}^t(\beta_1,\beta_2,\beta_3)_{(z)}.$$ A weight $2$ Bianchi modular form for $\Gamma$ can be alternatively described as a real analytic function $F=(F_1,F_2,F_3) : \mathcal{H}_3 \rightarrow \mathbb{C}^3$ such that $$F_1\beta_1 + F_2 \beta_2+F_3\beta_3$$ is a harmonic differential 1-form on $\mathcal{H}_3$ that is $\Gamma$-invariant. It is called cuspidal if it satisfies the extra property $$\int_{\mathbb{C} / \mathcal{O}_K} (F| \gamma )(x,y) dx = 0$$ for every $\gamma \in \textrm{PSL}_2(\mathcal{O}_K).$

This condition is equivalent to saying that the constant coefficient in the Fourier-Bessel expansion of $F|\gamma$ is equal to zero for every $\gamma \in \textrm{PSL}_2(\mathcal{O}_K)$.