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Equivalently, if $E/\\mathbb{C}(t)$ is a very general elliptic curve of height $d\\ge3$ and if $L$ is a finite extension of $\\mathbb{C}(t)$ with $L\\cong\\mathbb{C}(u)$, then the Mordell-Weil group $E(L)=0$."},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1407.7845","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.AG","submitted_at":"2014-07-29T19:47:20Z","cross_cats_sorted":[],"title_canon_sha256":"c959ff5d851b6a0a289c65a11dc112931b45964615cd9e2d475106fb1b954a98","abstract_canon_sha256":"2caf48aad7e326e86f93b7a618848153f46d4a3b516ec5bbd67f97b11abf39e2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:45:06.106766Z","signature_b64":"HzadrBi2dpEAK/9rZCqzBiezx2qoXrZyx/EqirMCsf4nzrdAzGu3cJF8qrpsCl1e56DXowPCyTCQzVV3P+5OAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"999f914d6bc9009183f1d688cd0e15480e2ed293a24974d46a644001fb23d268","last_reissued_at":"2026-05-18T02:45:06.106092Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:45:06.106092Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rational curves on elliptic surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AG","authors_text":"Douglas Ulmer","submitted_at":"2014-07-29T19:47:20Z","abstract_excerpt":"We prove that a very general elliptic surface $\\mathcal{E}\\to\\mathbb{P}^1$ over the complex numbers with a section and with geometric genus $p_g\\ge2$ contains no rational curves other than the section and components of singular fibers. 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