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Using more sophisticated methods, we then show that the Birch and Swinnerton-Dyer conjecture holds for E over K_d, and we relate the index of V_d in E(K_d) to the order of the Tate-Shafarevich group \\sha(E/K_d). When k has characteristic 0, we show that E has rank 0 over"},"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":"1002.3313","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2010-02-17T17:04:05Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"ac7e04cf0b0d084bf703ff9c237fb59d1a937d7a9c6613713dfccccdf863fe7b","abstract_canon_sha256":"a0cf2a93ac8069e7598aec5f0294d5f6d5b983a81591c3ebe45d328c4662028d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:12:48.807195Z","signature_b64":"A5CrPVl4pnEHCRHni7/fWsLxiWJmEzqpN0RrjhDQT4sAxlXXP0AA6Od9TudnZPYuJtzaK+JWVoQfKg6F0RXOBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"cb9f9598b7138db5cd413ae97895b9453eb23b4d88893ff7bb42a8cbc0704060","last_reissued_at":"2026-05-18T03:12:48.806536Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:12:48.806536Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Explicit points on the Legendre curve","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Douglas Ulmer","submitted_at":"2010-02-17T17:04:05Z","abstract_excerpt":"We study the elliptic curve E given by y^2=x(x+1)(x+t) over the rational function field k(t) and its extensions K_d=k(\\mu_d,t^{1/d}). 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