# The Doi-Naganuma Lift

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 $\tau$), where $\omega^{(k)}_m(z_1, z_2) = \sum_{(a,b,\lambda)}^{\prime} \frac{1}{(a z_1 z_2 + \lambda z_1 + \lambda^{\prime} z_2 + b)^k}$, the summation is over all triples $(a, b, \lambda)$ satisfying the conditions $a$, $b$ $\in$ $\mathbb{Z}$ and Norm of $\lambda$ - $ab$ = $\frac{m}{D}$, $\lambda$ $\in$ $\delta^{-1}$, $\delta$ is the principal ideal $(\sqrt{D})$. The notation $\lambda^{\prime}$ denote the conjugate of $\lambda$.

Zagier constructed $\Omega^{(k)}(z_1, z_2, \tau )$ (Reference: Modular forms associated to real quadratic fields"). The petersson inner product $\langle . , \Omega^{(k)} \rangle$ 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. My question is : What was the motivation for the construction of $\Omega^{(k)}(z_1, z_2, \tau )$? Thanks!

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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.) – David Loeffler Feb 12 '13 at 16:39
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). – Srilakshmi Feb 12 '13 at 17:03