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In this MathOverflow post, the smooth projective curve $C$ was defined over $\mathbb{C}$ and we have an isomorphism of de Rham cohomology groups $$H^1(C, \mathbb{C}) \cong H^1(J_C, \mathbb{C}),$$ where $J_C$ denotes the Jacobian variety of $C$. I believe such a result is also true if we were to consider curves over the algebraic closure of, say, a number field $k$. I am looking for a similar result in etale cohomology. More specifically, there are isomorphisms $$H^1_{et}(C,\mu_n) \cong H^1_{et}(X,\mu_n)$$ when $C$ is as above and $X$ is obtained by removing a $\bar{k}$-point of $C$, and $\mu_n$ is the subsheaf of $\mathbb{G}_m$.

Question. Let us suppose that the genus of $C$ is at least 2. Is it possible to construct an isomorphism between $H^1_{et}(C,\mu_n)$ and $H^1_{et}(J_C,\mu_n)$ where the varieties are defined over $\bar{k}$?

I have thought of using etale sheaf cohomology on the Kummer sequence and since $C$ and $J(C)$ are projective, I believe we will get an exact sequence $$0 \rightarrow H^1(C,\mu_n) \rightarrow \mathrm{Pic}(C) \rightarrow \mathrm{Pic}(C) \rightarrow H^2(C, \mu_n) \cong \mathbb{Z}/n\mathbb{Z} \rightarrow 0.$$ Here the last term is $0$ because it is the Brauer group of a curve over an algebraically closed field. If we do the same for $J_C$, I'm not sure how to go about computing $\mathrm{Pic}(J_C)$, $H^2(J_C,\mu_n)$ and of course its Brauer group.

Any help or references would be appreciated.

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    $\begingroup$ Pick a degree 1 divisor $D$ on $C$, consider the Abel-Jacobi map $\alpha _D:C\rightarrow J_C$, $\alpha _D(x)=[x]-D$, and take $\alpha _D^*$. THis is independent of the choice of $D$. $\endgroup$
    – abx
    Commented Oct 31, 2022 at 7:53
  • $\begingroup$ @abx I'm not exactly sure what $\alpha_D^*$ means, but I'm guessing it's the pull-back map sending a $J_C$-torsor $Y$ under $\mu_n$ to the $C$-torsor $Y\times_{J_C} C$? I believe such torsors are abelian coverings, then by applying Proposition 9 of Chapter 6 of Serre's Algebraic groups and class fields, we can show that the map $\alpha_D^*$ (if I defined it correctly) is surjective. I think injectivity is immediate for any map given by pull-backs. $\endgroup$
    – oleout
    Commented Oct 31, 2022 at 11:26

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