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Let $C$ be a smooth projective curve over a field $K$ of characteristic $0$ (but not necessarily algebraically closed). Let $\mathcal{L}$ be a line bundle on $C$ of degree $0$. Fix an integer $r>1$. Does there exist a line bundle $\mathcal{L}'$ on $C$ such that $\mathcal{L}'^{\otimes r} \cong \mathcal{L}$? If not true in general is there any condition on $K$ (like algebraic closedness) or $r$ under which this can hold true?

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No in general, for instance if $K$ is a number field : assuming that $C$ has a rational point, the group $\mathrm{Pic}(C)$ is naturally isomorphic to $JC(K)$, the group of rational points of the Jacobian of $C$, which is finitely generated by the Mordell-Weil theorem. So $\mathrm{Pic}(C)$ may very well contain points which are non-torsion and non-divisible by any $r$.

Yes if $K$ is algebraically closed : for any integer $r≠0$, multiplication by $r$ is a surjective endomorphism of $JC$, hence of $JC(K)$.

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  • $\begingroup$ Thank you very much for the answer. Could you please give a reference for the last statement (for the surjective endomorphism). $\endgroup$
    – Kali
    Commented Jul 20, 2014 at 17:55
  • $\begingroup$ Any introductory book on Abelian varieties; for instance Milne, ch. I, §7. $\endgroup$
    – abx
    Commented Jul 20, 2014 at 18:31

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