D\'{e}monstration g\'{e}om\'{e}trique du th\'{e}or\`{e}me de Lang-N\'{e}ron

We give a proof without heights of the Lang-N\'{e}ron theorem: if $K/k$ is a regular extension of finite type and $A$ is an abelian $K$-variety, the group $A(K)/\Tr_{K/k} A(k)$ is finitely generated, where $\Tr_{K/k} A$ denotes the $K/k$-trace of $A$ in the sense of Chow. Our method computes the rank of this group in terms of certain ranks of N\'{e}ron-Severi groups.