{"paper":{"title":"On the factorization of twisted $L$-values and $11$-descents over $C_5$-number fields","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"C\\'eline Maistret, Himanshu Shukla","submitted_at":"2025-01-16T13:02:01Z","abstract_excerpt":"We investigate the Galois module structure of the Tate-Shafarevich group of elliptic curves. For a Dirichlet character $\\chi$, we give an explicit conjecture relating the ideal factorization of $L(E,\\chi,1)$ to the Galois module structure of the Tate-Shafarevich group of $E/K$, where $\\chi$ factors through the Galois group of $K/\\mathbb{Q}$. We provide numerical evidence for this conjecture using the methods of visualization and $p$-descent. For the latter, we present a procedure that makes performing an $11$-descent over a $C_5$ number field practical for an elliptic curve $E/\\mathbb{Q}$ with"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2501.09515","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/2501.09515/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"}