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For all but finitely many $t\\in \\mathbb{Q}$, specialization will give an elliptic curve $E_t$ over $\\mathbb{Q}$ for which the abelian group $E_t(\\mathbb{Q})$ has rank at least $r$. Conjecturally, the set of $t\\in\\mathbb{Q}$ for which $E_t(\\mathbb{Q})$ has rank exactly $r$ has positive density. We produce the first known example for which $E_t(\\mathbb{Q})$ has rank $r$ for infinitely many $t\\in\\mathbb{Q}$. For our particular $E/\\mathbb{Q}(T)$ which has rank $0$, we "},"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":"2502.01026","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2025-02-03T03:44:34Z","cross_cats_sorted":[],"title_canon_sha256":"d5dc539d1dea29f6846cdf65f269cefeb2dd34b572efb184e09eecd05fc9d342","abstract_canon_sha256":"88ba62068c02d5077a892d4300c7e56b52f8eaaa3cbdae8b1c3458dbf98ddb48"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T10:08:41.803532Z","signature_b64":"m1Bu4I2J5FShe2nAfSrRMa02MpF6YLsD7ShMAARgqGHrLr56M0l5Z9EEAHJWqn4mArWvufTsQm/xckKpQlfTBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3fad564322e1dbe418d4351b0636c7e36870ee2aece54f219f0ff7537b935701","last_reissued_at":"2026-07-05T10:08:41.803111Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T10:08:41.803111Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"An elliptic surface with infinitely many fibers for which the rank does not jump","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"David Zywina","submitted_at":"2025-02-03T03:44:34Z","abstract_excerpt":"Let $E$ be a nonisotrivial elliptic curve over $\\mathbb{Q}(T)$ and denote the rank of the abelian group $E(\\mathbb{Q}(T))$ by $r$. For all but finitely many $t\\in \\mathbb{Q}$, specialization will give an elliptic curve $E_t$ over $\\mathbb{Q}$ for which the abelian group $E_t(\\mathbb{Q})$ has rank at least $r$. Conjecturally, the set of $t\\in\\mathbb{Q}$ for which $E_t(\\mathbb{Q})$ has rank exactly $r$ has positive density. We produce the first known example for which $E_t(\\mathbb{Q})$ has rank $r$ for infinitely many $t\\in\\mathbb{Q}$. 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