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Let $C$ be a curve and $J$ be its Jacobian.

What is the relation between $H^1(C,\mathcal{O}_C)$ and $H^1(J,\mathcal{O}_J)$ ?

Can someone point me to an easy reference for this subject?

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They are the same in characteristic 0, because the Abel-Jacobi map induces an isomorphism $ H^0(C, \Omega) \to H^0(J, \Omega) $ and (the notation below means singular cohomology with complex coefficients of the associated complex manifold) it also induces an isomorphism $ H^1(C) \to H^1(J) $ but we have (canonically, but not naturally, split) exact sequences (for any $X$) $ 0 \to H^0(X, \Omega) \to H^1(X) \to H^1(X, \mathcal{O}) \to 0. $

In characteristic zero you can find all this in Birkenhakke and Lange's book on Complex Abelian Varieties.

I believe the result is true for basically the same reason using an appropriate Weil-type cohomology theory in characteristic $p$ or a lifting argument. Perhaps there is an easier argument. I don't know a reference in characteristic $p$. You might try Milne's notes on his website.

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    My argument is dead wrong in characteristic $p$ - the fields of coefficients of the cohomology groups don't agree! However, I believe the claim is still true using a lifting argument to characteristic 0, but you will need more details from someone else. Editing response.2012-04-25