# Roots of a canonical line bundle on a compact Riemann surface

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?

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