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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 $\gamma \in \textrm{PSL}_2(\mathcal{O}_K)$.

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  • Review status: beta
  • Last edited by Holly Swisher on 2019-05-08 14:04:34
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