Existence of a line bundle of order 2

Question

For a genus $g$ (nonsingular irreducible) curve, does there always exist a line bundle $L$ such that $L^{\otimes2}=O$ (besides the trivial one)? For genus 0, this is an easy question by every line bundle is isomorphic to $O(m)$ and thus only trivial one.

Answer 1

Assuming the curve is of positive genus and over an algebraically closed field $k$ where $2$ is invertible, the answer is yes. In the case where $2$ is not invertible, things may go wrong. See https://stacks.math.columbia.edu/tag/0C1Y for a reference - the key ingredients of the proof are that degree-zero invertible sheaves on a projective curve are classified by an abelian variety, and that we know what torsion looks like in an abelian variety.