{"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 $t\\in\\mathbb{Q}$. For our particular $E/\\mathbb{Q}(T)$ which has rank $0$, we "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2502.01026","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2502.01026/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}