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 $D$?



Explicit formula for the Poincare dual of a CM endomorphism of an elliptic curve
Among $E\_1, E\_2, \Gamma, \Delta$ we have the intersections:


$E\_1 \cdot E\_1 = E\_2\cdot E\_2 = \Delta \cdot \Delta =\Gamma \cdot \Gamma =0 $


$E\_1 \cdot E\_2 = E\_1 \cdot \Delta = E\_2 \cdot \Delta =1$


$E\_1 \cdot \Gamma = 1$


$E\_2 \cdot \Gamma = D$


$\Delta \cdot \Gamma = 1+D$.


Presumably only the last computation will need explanation. The reason is just that you are coutining solutions to $(1-\sqrt{-D}) (x) \in \mathbb Z[\tau]$ modulo $\mathbb Z[\tau]$. The number of solutions is always the norm of $1-\sqrt{-D}$, or $1+D$.


Does this help?


