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I can't find anywhere a proof of the following comparison isomorphishm:

$$H^1_{dR}(E)\otimes \mathbb{C}=H^1_{et}(E)\otimes \mathbb{C}$$

where $E$ is an elliptic curve over $\mathbb{C}$.

Any reference would be appreciated.

On a side note, is the isomorphism true for $n>1$?

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  • $\begingroup$ I think you want $\otimes$ rather than $\oplus$. This is étale cohomology with coefficients in what? $\endgroup$ Sep 9, 2015 at 0:42
  • $\begingroup$ @BrunoJoyal Of course you are right. Not sure about the coefficients, I've seen this stated without much explanation (only the p-adic analogy, with the ring of periods substituting $\mathbb{C}$). $\endgroup$ Sep 9, 2015 at 0:50
  • $\begingroup$ Well, what do you think the right statement should be? $\endgroup$ Sep 9, 2015 at 15:23

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