{"paper":{"title":"Elliptic curves and Hilbert's tenth problem for algebraic function fields over real and p-adic fields","license":"","headline":"","cross_cats":["math.AG"],"primary_cat":"math.LO","authors_text":"Laurent Moret-Bailly","submitted_at":"2004-09-07T12:47:03Z","abstract_excerpt":"Let k be a field of characteristic zero, V a smooth, positive-dimensional, quasiprojective variety over k, and D a nonempty effective divisor on V. Let K be the function field of V, and A the semilocal ring of D in K.\n In this paper, we prove the Diophantine undecidability of:\n (1) A, in all cases;\n (2) K, when k is (formally) real and V has a real point;\n (3) K, when k is a subfield of a p-adic field, for some odd prime p.\n To achieve this, we use Denef's method: from an elliptic curve E over Q, without complex multiplication, one constructs a quadratic twist E' of E over Q(t), which has Mord"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"math/0409103","kind":"arxiv","version":3},"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"}