Let $K$ be a number field, and let $C/K$ be a curve of genus $g \geq 2$ over $K$. Suppose that the set of $K$-rational points $C(K)$ on $C$ is non-empty. This enables one to define an Abel-Jacobi map $\sigma$ defined over $K$, with a given point $P \in C(K)$, which gives an embedding of $C$ into the Jacobian $J_C$, i.e., $\sigma : C \hookrightarrow J_C$.

Let $J_C(K)$ be the Mordell-Weil group of $J_C$ over $K$, and let $J_C(K)^{\text{tors}}$ be the set of points of finite order. What is the pre-image of $J_C(K)^{\text{tors}}$ in $C$ under $\sigma$?

Are there non-trivial examples (i.e., when $C(K) = \emptyset$ or $|J_C(K)^{\text{tors}}| = 1$) of when $C(K) = \sigma^{-1} \left(J_C(K)^{\text{tors}}\right)$?

Let $K$ be a number field, and let $C/K$ be a curve of genus $g \geq 2$ over $K$. Suppose that the set of $K$-rational points $C(K)$ on $C$ is non-empty. This enables one to define an Abel-Jacobi map $\sigma$ defined over $K$, with a given point $P \in C(K)$, which gives an embedding of $C$ into the Jacobian $J_C$, i.e., $\sigma : C \hookrightarrow J_C$.

Let $J_C(K)$ be the Mordell-Weil group of $J_C$ over $K$, and let $J_C(K)^{\text{tors}}$ be the set of points of finite order. What is the pre-image of $J_C(K)^{\text{tors}}$ in $C$ under $\sigma$?

Are there non-trivial examples (i.e., when $C(K) \ne \emptyset$ and $|J_C(K)^{\text{tors}}| > 1$) of when $C(K) = \sigma^{-1} \left(J_C(K)^{\text{tors}}\right)$?