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A similar theorem is valid in case when $\\op{deg}f=4$ and there exists $t_{0}\\in\\Q$ such that infinitely many rational points lie on the curve $E_{t_{0}}:y^2=x^3+f(t_{0})x$. In particular, we prove that if $\\op{deg}f=4$ and $f$ is not an even polynomial, then there is a rational point on $\\cal{E}_{f}$"},"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":"0705.2955","kind":"arxiv","version":1},"metadata":{"license":"","primary_cat":"math.NT","submitted_at":"2007-05-21T12:10:43Z","cross_cats_sorted":[],"title_canon_sha256":"97fd72bf72d8ede5a0fe03cd652d3af9244b76a31d40883d1ec8f87f1b98ac90","abstract_canon_sha256":"907f11d93834225eb0d76fd4094f12f5ba3a64658d1caaad5e2fc0e23b9babda"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:16:30.792885Z","signature_b64":"3tH5meMekYov9IDxqr5J/hZw3aGckAouV5sjuFncHb+o4oyvt04a3bEHElG2et/e80TJpsamEoAnWcWtiOi/Cw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"073acf276e5c987f49117e2c86a00663a02c47af99de1638ec61a7ee24778710","last_reissued_at":"2026-05-18T02:16:30.792310Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:16:30.792310Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Rational points on certain elliptic surfaces","license":"","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Maciej Ulas","submitted_at":"2007-05-21T12:10:43Z","abstract_excerpt":"Let $\\mathcal{E}_{f}:y^2=x^3+f(t)x$, where $f\\in\\Q[t]\\setminus\\Q$, and let us assume that $\\op{deg}f\\leq 4$. In this paper we prove that if $\\op{deg}f\\leq 3$, then there exists a rational base change $t\\mapsto\\phi(t)$ such that on the surface $\\cal{E}_{f\\circ\\phi}$ there is a non-torsion section. A similar theorem is valid in case when $\\op{deg}f=4$ and there exists $t_{0}\\in\\Q$ such that infinitely many rational points lie on the curve $E_{t_{0}}:y^2=x^3+f(t_{0})x$. 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