Let $C$ be a curve (=smooth projective curve) of genus $g$ over an algebraic closed field $\mathbb{k}$. It is well known that the Grothendieck group $K_0(\operatorname{coh} C)$ of the category of coherent sheaves on $C$ is the following (see, for example, Exercise 6.11 of Hartshorne): $$ K_0(\operatorname{coh} C)\simeq \operatorname{Pic} C \times \mathbb{Z} \simeq \operatorname{Pic}_0 C \times \mathbb{Z}^2, $$ where $\operatorname{Pic} C$ is the Picard group of $C$ and $\operatorname{Pic}_0 C$ is the kernel of the degree map $\mathrm{deg} : \operatorname{Pic} C\to \mathbb{Z}$. In the case $\mathbb{k} = \mathbb{C}$, we also know that $$ \operatorname{Pic}_0 C \simeq \mathbb{C}^g/\Lambda, $$ where $\Lambda$ is a $\mathbb{Z}$-submodule in $\mathbb{C}^g$ of rank $2g$.

**My Questions:**

- For two curves $C_1$ and $C_2$, if $K_0(\operatorname{coh} C_1) \simeq K_0(\operatorname{coh} C_2)$ as abstract groups,
then do we conclude that $\operatorname{Pic}_0 C_1 \simeq \operatorname{Pic}_0 C_2$
(or $\operatorname{Pic} C_1 \simeq \operatorname{Pic} C_2$)
as abstract groups ?
Here
*abstract groups*means groups without additional structure, such as structure of topology, variety and so on. - Does an isomorphism $\operatorname{Pic}_0 C_1 \simeq \operatorname{Pic}_0 C_2$ of abstract groups contain some geometric information about $C_1$ and $C_2$?