Revisions to Explicit formula for the Poincare dual of a CM endomorphism of an elliptic curve

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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}/\mathfrak{a}$$E=\mathbf{C}/\Lambda_{\tau}$, where $\mathfrak{a}\subseteq \mathcal{O}_K$$\Lambda_{\tau}=\mathbf{Z}+\tau\mathbf{Z}$ and $\tau$ is a fixed integral idealin 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 orientation induced metric coming from the orientation of $E$ 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: if one thinks of the torus as a rectangle with the usual side idtentifications, then $E$: $a$ corresponds to cohomology class of the horizontaloriented vector joining (left$0$ to right) edge$1$, $b$ corresponds to the verticalcohomology class of the oriented vector joining (bottom$0$ to top) edge and $\tau$, so $a\cdot b=1$ (soor $b\cdot a=-1$). 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 ideal callPicard class of $\mathfrak{a}$$\Lambda_{\tau}$ in general.

Q: Is there a method to compute thesethe numbers $c_i$'s in a systematic way, so that we can clearly see their dependence on $D$?

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 $E=\mathbf{C}/\mathfrak{a}$ where $\mathfrak{a}\subseteq \mathcal{O}_K$ is a fixed integral ideal. We endow $E$ with the usual flat metric coming from the complex plane. 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 orientation induced from the orientation of $E$ via the map $[\sqrt{-D}]$. 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: if one thinks of the torus as a rectangle with the usual side idtentifications, then $a$ corresponds to the horizontal (left to right) edge, $b$ to the vertical (bottom to top) edge and $a\cdot b=1$ (so $b\cdot a=-1$). 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 ideal call of $\mathfrak{a}$ in general.

Q: Is there a method to compute these numbers in a systematic way, so that we can clearly see their dependence on $D$?

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.