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Let $F$=$\mathbb{Q}(\sqrt{D})$ be a real quadratic field and $\mathcal{O}_F$ be the ring of integers of $F$. The generating series $$\Omega^{(k)}(z_1, z_2, \tau ) := \sum^{\infty}_{m=1} m^{k-1} \omega^{(k)}_m(z_1, z_2) e^{2\pi i m \tau}$$ ($z_1$, $z_2$, $\tau$ $\in$ $\mathbb{H}$, the upper half plane) is both a Hilbert modular form (with respect to $z_1$ and $z_2$) and a classical modular form (with respect to $\tau$). Here $$\omega^{(k)}_m(z_1, z_2) := \sum_{\scriptstyle{a,b\in \mathbb{Z},\ \ \lambda \in \delta^{-1},}\atop\scriptstyle{N(\lambda)-ab = \frac{m}{D}} } \frac{1}{(a z_1 z_2 + \lambda z_1 + \lambda^{\prime} z_2 + b)^k},$$ where $\delta$ is the principal ideal $(\sqrt{D})$ and $\lambda^{\prime}$ denotes the conjugate of $\lambda$.

$\Omega^{(k)}(z_1, z_2, \tau )$ has been constructed by Zagier (reference: "Modular forms associated to real quadratic fields"). The Petersson inner product $\langle \cdot , \Omega^{(k)} \rangle$ defines a linear map from classical cusp forms to Hilbert cusp forms, $S_k(\Gamma_0(D), \chi_D) \to S_{(k,k)}(SL_2(\mathcal{O}_F))$.

I'd like to know the history behind that. My question is :

What was the motivation for the construction of $\Omega^{(k)}(z_1, z_2, \tau )$?

Thanks!

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  • $\begingroup$ This is a nice question, IMHO, but it has nothing to do with arithmetic geometry as far as I can see. (I removed the tag and you seem to have added it back.) $\endgroup$ – David Loeffler Feb 12 '13 at 16:39
  • $\begingroup$ Thanks for removing the tag. There is a geometric interpretation of the Doi-Naganuma lifting and it's adjoint. I amn't sure arithmetic-geometry tag would be suitable (Probably you can retag). $\endgroup$ – Srilakshmi Feb 12 '13 at 17:03

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