Algebraic curves and torsion points of its Jacobian 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 $C(K) = \sigma^{-1} \left(J_C(K)^{\text{tors}}\right)$?
 A: Trivially (though non-trivially according to your definition), we have that $C(K) = \sigma^{-1}(J_C(K)^{\text{tors}})$ whenever $J_C(K)$ is finite. There are lots of examples of this kind. But it also should not be very hard to come up with examples of, say, curves of genus 2 over $\mathbb Q$ whose Jacobian has positive rank (and non-trivial torsion), but such that no point of infinite order can be written as a difference of two rational points on the curve: Searching the LMFDB for genus 2 curves with a single rational point whose Jacobian has positive rank and nontrivial torsion turns up a fairly long list of examples, the first of which is this curve:
$$ y^2 + y = x^5 + 4x^3 - 6x^2 + x - 15 $$
that has the unique point at infinity as its only rational point and Mordell-Weil group isomorphic to ${\mathbb Z} \times {\mathbb Z}/3{\mathbb Z}$.
There certainly will also be examples with more than one rational point on the curve (e.g., when the curve is hyperelliptic and all rational points are Weierstrass points).
