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Suppose we have a compact Riemann surface $X$ of genus $g$. Let $K$ denote the canonical line bundle on $X$, it's well known that $deg\ K=2g-2$. A square root of $K$ by definition is a holomorphic line $L$ bundle that satisfies $L\otimes L=K$. Then are $2^{2g}$ different square roots, e.g. see this link .

Now suppose $m$ is an integer such that $m$ divides $deg\ K=2g-2$. Let's call a holomorphic line bundle $L$ for which $L^{\otimes m}=K$, an $m$-root of $K$. Is true in general that they are $m^{2g}$ different holomorphic $m$-roots?

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    $\begingroup$ Yes, the number of different n-roots are just the number of n-torsion points of its albanese variety. $\endgroup$
    – Chen Jiang
    Feb 25, 2017 at 3:36
  • $\begingroup$ @ChenJiang Could you give some references, please? $\endgroup$
    – yaa09d
    Feb 26, 2017 at 5:25

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