Let $X$ a Riemann surface of genus $g$. How could I prove that there is an isomorphism between $Pic_X[2]$ (i.e the line bundles $L$ such as $L^2 \simeq \mathcal{O}_X $) and $(\mathbb{Z}/2\mathbb{Z})^{2g}$?
I know that, because of the Weil's Reciprocity Law, we can define the following Weil pairing
$Pic_X[2] \times Pic_X[2] \to \mathbb{Z}/2\mathbb{Z}$,
but I don't know if it could be useful.