# 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 $$\tau$$). 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!

• 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