{"paper":{"title":"On the Minimality of the Conductor for Elliptic Curve $L$-Functions","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"K. Lakshmanan","submitted_at":"2025-06-25T07:03:00Z","abstract_excerpt":"We investigate the role of the conductor in analytic rank bounds for elliptic curves over \\(\\mathbb{Q}\\). Let \\(E/\\mathbb{Q}\\) be an elliptic curve with conductor \\(N_E\\). We consider hypothetical degree-two \\(L\\)-functions associated to (E) that satisfy analytic continuation, a functional equation involving an arithmetic invariant \\(\\Phi(E)\\), and yield rank bounds of the form\n  \\[\n  \\operatorname{rank}(E)\\ll \\log \\Phi(E).\n  \\]\n  Using the Modularity Theorem, we show that any such invariant must satisfy\n  \\[\n  \\Phi(E)\\ge N_E.\n  \\] Thus the conductor is minimal among arithmetic invariants that"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2506.20175","kind":"arxiv","version":2},"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/2506.20175/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"}