A curve and its abstract generalized Jacobian

To a smooth proper curve $C$ over a field $k$ equipped with a $k$-point $c$ and an effective divisor $\mathfrak m$ coprime to $c$, one may associate the abstract group $J_{\mathfrak m}(\bar k)$ of $\overline k$-points of the generalized Jacobian, as well as a subset $$\tag{*} \big(C\setminus \operatorname{Supp}(\mathfrak m)\big)(\bar k) \subset J_{\mathfrak m}(\bar k).$$ We show that the data $(C,c,\mathfrak m)$ can be retrieved from (*) up to a twist by an automorphism of $\overline k$, proving a conjecture of Booher and Voloch. By a result of Booher and Voloch this shows that when $k$ is a finite field, the same data may also be retrieved from $L$-functions of characters of certain Galois extensions of the function field of $C$. The proof is a generalization of Zilber's well known work "A curve and its abstract Jacobian".