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I'd like to find a reference to the following statement.

Let $k$ be a separably closed field of characteristic $p$, and $\phi: C_1\to C_2$ be a morphism of smooth projective curves over $k$. Assume that $\deg \phi$ is coprime to $p$. Let $\phi^*: J_2\to J_1$ be the morphism of the Jacobian varieties induced by the Picard functoriality, and $K$ be the kernel of $\phi^*$. Then the $\mathbb{G}_m$-dual $K^\vee$ of $K$ is isomorphic to covering group of the maximal abelian unramified covering of $C_2$ which is intermediate to $\phi: C_1\to C_2$.

This result (at least in characteristic 0) is probably well known to the experts, since it is used without any reference in a paper. Hopefully, I have formulated it correctly for characteristic $p$. I'd like to use it in my own paper, but I don't know where to find it in the literature.

Thank you.

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1  
Serre "Groupes algebriques et corps de classes". –  Felipe Voloch Jul 23 '12 at 12:05
2  
Serre's book assumes $k$ is algebraically closed, but it is an instructive exercise in Galois theory (applied to the Galois closure of $k(C_1)$ over $k(C_2)$) to reduce to that case (using that $\overline{k}$ is a purely inseparable extension of $k$). Also note that by the duality theory of abelian varieties (and the dual relationship between Picard and Albanese functoriality for curves), $K^{\vee}$ is the kernel of the maximal isogenous cover $f:A \rightarrow J_2$ intermediate to the Albanese map $J_1 \rightarrow J_2$ dual to $\phi^{\ast}$. Apply Serre's book to $f$. –  user22479 Jul 23 '12 at 13:41

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