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Let $F$ be a Bianchi modular form with respect to $\Gamma$. 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 variable $x=(x_1,x_2)$. 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.

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• Review status: beta
• Last edited by Holly Swisher on 2019-05-08 13:56:26
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