Lemma 53.17.1. Let $k$ be an algebraically closed field. Let $X$ be a smooth projective curve of genus $g$ over $k$.

If $n \geq 1$ is invertible in $k$, then $\mathop{\mathrm{Pic}}\nolimits (X)[n] \cong (\mathbf{Z}/n\mathbf{Z})^{\oplus 2g}$.

If the characteristic of $k$ is $p > 0$, then there exists an integer $0 \leq f \leq g$ such that $\mathop{\mathrm{Pic}}\nolimits (X)[p^ m] \cong (\mathbf{Z}/p^ m\mathbf{Z})^{\oplus f}$ for all $m \geq 1$.

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