{"paper":{"title":"D\\'{e}monstration g\\'{e}om\\'{e}trique du th\\'{e}or\\`{e}me de Lang-N\\'{e}ron","license":"","headline":"","cross_cats":["math.NT"],"primary_cat":"math.AG","authors_text":"Bruno Kahn (IMJ)","submitted_at":"2007-03-02T15:50:57Z","abstract_excerpt":"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."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0703063","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}