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Let $E$ be an elliptic curve over a field $k$. Let

$$\mu:E \times_k E \to E$$

be the addition map on $E$. Furthermore let $p_1,p_2:E \times_k E \to E$ be the two canonical projections and let ${\mathcal L} = \mathcal{L}(D)$ be the line bundle of a divisor $D$ with $\deg D = 0$ on $E$, that is

$$\mathcal{L} \in \mathrm{Pic}^0(E)$$

Is it true, that

$$(*) \quad \mu^* \mathcal{L} = p_1^*\mathcal{L} \otimes p_2^*\mathcal{L}$$

I am "90 percent sure" to have a proof for this (using the semicontinuity theorem) but a certain doubt remains, as I could find the result neither in Hartshorne's "Algebraic geometry" chapter on curves, nor in Silverman's "The arithmetic of elliptic curves", although it looks like a fundamental fact (if true). (Explanation: I try to use $(*)$ to prove

$$\hat{(\phi + \psi)} = \hat{\phi} + \hat{\psi}$$

for two isogenies $\phi, \psi: E_1 \to E_2$ of elliptic curves, where $\hat{\phi}$ means the dual isogeny to $\phi$).

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    $\begingroup$ I finally found out the following: By Mumford, "Abelian Varieties", 1970, p. 74,75 the contended assertion $(*)$ is even true for a general abelian variety $X$ instead of $E$. $\endgroup$ Apr 16, 2016 at 22:05

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