Let $E/\mathbf{C}$ be an elliptic curve with CM by the maximal order $\mathcal{O}_K$ of $K=\mathbf{Q}(\sqrt{-D})$ where $D$ is positive and square-free integer. To make it even more precise, let us assume that $E=\mathbf{C}/\Lambda_{\tau}$, where $\Lambda_{\tau}=\mathbf{Z}+\tau\mathbf{Z}$ and $\tau$ is in the Poincare upper half-plane. We endow $E$ with the usual flat metric coming from the complex plane, so $E$ comes equipped with a volume form. Consider the endomorphism $[\sqrt{-D}]:E\rightarrow E$. Let $\Gamma\subseteq E\times E$ be the graph of $[\sqrt{-D}]$. Then $\Gamma$ is isomorphic (as a complex manifold) to $E$, in particular it is a 2-cycle on $E\times E$. We endow $\Gamma$ with the induced metric coming from $E$ via the map $[\sqrt{-D}]$. We also endow $E\times E$ with the product metric coming from $E$. Now let $\eta$ be the Poincare dual of $\Gamma$: so if $\iota:\Gamma\hookrightarrow E\times E$ denotes the inclusion then for all $\omega\in A^2(E\times E)$, we have $\int_{\Gamma}\iota^*\eta=\int_{E\times E}\omega\wedge \eta$. Let $a,b\in H^1(E,\mathbf{Z})$ be the standard oriented basis of $E$: $a$ corresponds to cohomology class of the oriented vector joining $0$ to $1$, $b$ corresponds to the cohomology class of the oriented vector joining $0$ to $\tau$, so $a\cdot b=1$ (or $b\cdot a=-1$). From the Kuenneth formula we have a priviledged isomorphism $H^*(E\times E)\simeq\oplus_{i=1}^4 H^{i}(E)\otimes H^{4-i}(E)$. Via this isomorphism, there exists real numbers $c_i$ ($i=1\ldots 4$) such that $$ \eta=c_1\cdot(a\otimes a)+c_2\cdot(a\otimes b)+c_3\cdot(b\otimes a)+c_4\cdot(b\otimes b). $$ I would expect in general the $c_i$'s to depend on $D$. On the other hand I don't know if the $c_i$'s will depend on the Picard class of $\Lambda_{\tau}$ in general.
Q: Is there a method to compute the numbers $c_i$'s in a systematic way, so that we can clearly see their dependence on $D$?
