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We also proved the finiteness of $III(E/K_d)$ and a class number formula: $[E(K_d):V_d]^2=|III(E/K_d)|$. In this paper, we compute $E(K_d)/V_d$ and $III(E/K_d)$ explicitly as modules over $\\mathbf{Z}_p[\\mathrm{Gal}(K_d/F_p(t))]$."},"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":"1406.6674","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2014-06-25T19:26:55Z","cross_cats_sorted":[],"title_canon_sha256":"37fbcb399b579da239e28f5b2e1d5e2120ffbc8dc8bf544434357ef4b465131d","abstract_canon_sha256":"5b094db308d1d95b0d1c71e377eb9ee0a43272c4be75fa58759fdb2346624e89"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:43:45.609208Z","signature_b64":"27PLxeRfwfCtKXYrqnSR5xwVTXtqg64KposXbSFzulGoNUQcwxkKs3jVkwDOF3M3LWmrrf7UJp5VJM8F40V3BA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"be65d642975244f0d3d28612bc2138b0cbd75b16268ecd5fd95309c3477d31a2","last_reissued_at":"2026-05-18T00:43:45.608737Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:43:45.608737Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Explicit points on the Legendre curve III","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Douglas Ulmer","submitted_at":"2014-06-25T19:26:55Z","abstract_excerpt":"We continue our study of the Legendre elliptic curve $y^2=x(x+1)(x+t)$ over function fields $K_d=\\mathbf{F}_p(\\mu_d,t^{1/d})$. When $d=p^f+1$, we have previously exhibited explicit points generating a subgroup $V_d$ of $E(K_d)$ of rank $d-2$ and of finite, $p$-power index. We also proved the finiteness of $III(E/K_d)$ and a class number formula: $[E(K_d):V_d]^2=|III(E/K_d)|$. In this paper, we compute $E(K_d)/V_d$ and $III(E/K_d)$ explicitly as modules over $\\mathbf{Z}_p[\\mathrm{Gal}(K_d/F_p(t))]$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.6674","kind":"arxiv","version":3},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1406.6674","created_at":"2026-05-18T00:43:45.608820+00:00"},{"alias_kind":"arxiv_version","alias_value":"1406.6674v3","created_at":"2026-05-18T00:43:45.608820+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1406.6674","created_at":"2026-05-18T00:43:45.608820+00:00"},{"alias_kind":"pith_short_12","alias_value":"XZS5MQUXKJCP","created_at":"2026-05-18T12:28:57.508820+00:00"},{"alias_kind":"pith_short_16","alias_value":"XZS5MQUXKJCPBU6S","created_at":"2026-05-18T12:28:57.508820+00:00"},{"alias_kind":"pith_short_8","alias_value":"XZS5MQUX","created_at":"2026-05-18T12:28:57.508820+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/XZS5MQUXKJCPBU6SQYJLYIJYWD","json":"https://pith.science/pith/XZS5MQUXKJCPBU6SQYJLYIJYWD.json","graph_json":"https://pith.science/api/pith-number/XZS5MQUXKJCPBU6SQYJLYIJYWD/graph.json","events_json":"https://pith.science/api/pith-number/XZS5MQUXKJCPBU6SQYJLYIJYWD/events.json","paper":"https://pith.science/paper/XZS5MQUX"},"agent_actions":{"view_html":"https://pith.science/pith/XZS5MQUXKJCPBU6SQYJLYIJYWD","download_json":"https://pith.science/pith/XZS5MQUXKJCPBU6SQYJLYIJYWD.json","view_paper":"https://pith.science/paper/XZS5MQUX","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1406.6674&json=true","fetch_graph":"https://pith.science/api/pith-number/XZS5MQUXKJCPBU6SQYJLYIJYWD/graph.json","fetch_events":"https://pith.science/api/pith-number/XZS5MQUXKJCPBU6SQYJLYIJYWD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/XZS5MQUXKJCPBU6SQYJLYIJYWD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/XZS5MQUXKJCPBU6SQYJLYIJYWD/action/storage_attestation","attest_author":"https://pith.science/pith/XZS5MQUXKJCPBU6SQYJLYIJYWD/action/author_attestation","sign_citation":"https://pith.science/pith/XZS5MQUXKJCPBU6SQYJLYIJYWD/action/citation_signature","submit_replication":"https://pith.science/pith/XZS5MQUXKJCPBU6SQYJLYIJYWD/action/replication_record"}},"created_at":"2026-05-18T00:43:45.608820+00:00","updated_at":"2026-05-18T00:43:45.608820+00:00"}