Why is the Picard group of a smooth projective curve over a number field finitely generated?

Question

I've been trying to understand a paper which basically states that the Picard group of a smooth projective curve over a number field is finitely generated. The only thing I found on the internet is an answer on this site supporting this assertion without proof. I'm looking for a proof ever since without success.

Furthermore, the paper claims that this follows from the Mordell-Weil Theorem, but isn't the Mordell-Weil Theorem about abelian varieties and rational points? I don't see any connection.

Answer 1

Consider a smooth projective curve $C$ of genus $\geq 1$ over a number field $K$. Since we have an exact sequence

$$ 0 \rightarrow \text{Pic}^0(C) \rightarrow \text{Pic}(C) \rightarrow \text{NS}(C) \rightarrow 0,$$

it suffices to show that $\text{Pic}^0(C)$ and $\text{NS}(C)$ are finitely generated.

A famous result says that the Néron-Severi group, $\text{NS}(C)$, is finitely generated.

Being an abelian variety over a number field, $\text{Pic}^0(C)$ is finitely generated by Mordell-Weil.

Answer 2

Actually there is a generalization of the Mordell-Weil Theorem, which is as follows:

Theorem (Mordell-Weil-Lang-Neron) Let $K$ be a field that is of finite type over its prime field (where the prime field is either $\mathbb Q$ or $\mathbb F_p$), and let $A/K$ be an abelian variety. Then $A(K)$ is finitely generated.

The proof can be found in Lang's Fundamentals of Diophantine Geometry. We can chose $K$ as a number field to obtain the Mordell-Weil theroem for an abelian variety $A$ over a number field $K$. This yields the claim concerning the Picard group via the Jacobian, see here at MO. There also is a discussion that for some fields $Pic^0(X)$ may not be finitely-generated.