{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:NQ3F5D7GRUWL3UAYEHQACO5E73","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"e5d6ae7f44df76dd7884b8b51a9c21035b80fca80596e0134b0f29d6cf20b07c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-02-08T08:42:41Z","title_canon_sha256":"ddf6a2f02f15d957857b06cc516d5fba58e03714e2c8a88202fad06262fe6a57"},"schema_version":"1.0","source":{"id":"1702.02325","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1702.02325","created_at":"2026-05-18T00:42:51Z"},{"alias_kind":"arxiv_version","alias_value":"1702.02325v2","created_at":"2026-05-18T00:42:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1702.02325","created_at":"2026-05-18T00:42:51Z"},{"alias_kind":"pith_short_12","alias_value":"NQ3F5D7GRUWL","created_at":"2026-05-18T12:31:34Z"},{"alias_kind":"pith_short_16","alias_value":"NQ3F5D7GRUWL3UAY","created_at":"2026-05-18T12:31:34Z"},{"alias_kind":"pith_short_8","alias_value":"NQ3F5D7G","created_at":"2026-05-18T12:31:34Z"}],"graph_snapshots":[{"event_id":"sha256:4be160288217fd1d905389fdccdbe61feb84442e525dd2f68f73f2a7e8faa7c0","target":"graph","created_at":"2026-05-18T00:42:51Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"We prove that the $2^\\infty$-class groups of the imaginary quadratic fields have the distribution predicted by the Cohen-Lenstra heuristic. Given an elliptic curve E/Q with full rational 2-torsion and no rational cyclic subgroup of order four, we analogously prove that the $2^\\infty$-Selmer groups of the quadratic twists of E have distribution as predicted by Delaunay's heuristic. In particular, among the twists E^d with |d| < N, the number of curves with rank at least two is $o(N)$.","authors_text":"Alexander Smith","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-02-08T08:42:41Z","title":"$2^\\infty$-Selmer groups, $2^\\infty$-class groups, and Goldfeld's conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1702.02325","kind":"arxiv","version":2},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:864157ad6013657399e56797a004e6e4caa2c7976f39d92e95bc77168b76b8fd","target":"record","created_at":"2026-05-18T00:42:51Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"e5d6ae7f44df76dd7884b8b51a9c21035b80fca80596e0134b0f29d6cf20b07c","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-02-08T08:42:41Z","title_canon_sha256":"ddf6a2f02f15d957857b06cc516d5fba58e03714e2c8a88202fad06262fe6a57"},"schema_version":"1.0","source":{"id":"1702.02325","kind":"arxiv","version":2}},"canonical_sha256":"6c365e8fe68d2cbdd01821e0013ba4fec7d65311407692f42796650541696429","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6c365e8fe68d2cbdd01821e0013ba4fec7d65311407692f42796650541696429","first_computed_at":"2026-05-18T00:42:51.293222Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:42:51.293222Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"d3QV6+KS8rWTkE9i9LYyunzqcW+v4qP40ShYDCjo0aW9DXlSCgriJvIkNHfU5znCKy99IyHYrtC8fVZDDBzRAw==","signature_status":"signed_v1","signed_at":"2026-05-18T00:42:51.293948Z","signed_message":"canonical_sha256_bytes"},"source_id":"1702.02325","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:864157ad6013657399e56797a004e6e4caa2c7976f39d92e95bc77168b76b8fd","sha256:4be160288217fd1d905389fdccdbe61feb84442e525dd2f68f73f2a7e8faa7c0"],"state_sha256":"2d1fb14856f6e49e129d88ebdc739486b60329600932b3a3799212948ebe6713"}