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If $E_1, E_2$ are elliptic curves over $\mathbb{C}$ associated to torus $\mathbb{C} / \Lambda_1, \mathbb{C} / \Lambda_2$, an isogeny $\Phi: E_1 \to E_2$ is determined by an $\alpha \in \mathbb{C}$ such that $\alpha \Lambda_1 \subset \Lambda_2$ and corresponds to the map $z \mapsto \alpha z \mod \Lambda_2$ (see Silvermann VI.4.1).

Given that interpretation of isogenies as scalar multiplication, how can we interpret the existence of the dual isogeny? (an application $\hat{\Phi}: E_2 \to E_1$ such that $\hat{\Phi} \circ \Phi = [m]$ where $m$ is the degree of $\Phi$).

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The fact that the degree of $E_1 \to E_2$ is $m$ implies that $m \Lambda_2 \subset \alpha \Lambda_1$. Rescaling this gives the dual isogeny.

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  • $\begingroup$ What is the interpretation of the degree when considering an isogeny as scalar multiplication on the torus? $\endgroup$
    – Weier
    Jul 4, 2023 at 11:19
  • $\begingroup$ Could you please detail a bit your answer? Why do we have this implication? I would accept an answer with more details. $\endgroup$
    – Weier
    Jul 4, 2023 at 16:21

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