{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:S7YUTCGLRLK55VOLNYBMJWIQBS","short_pith_number":"pith:S7YUTCGL","canonical_record":{"source":{"id":"1603.01296","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-03-03T21:59:05Z","cross_cats_sorted":[],"title_canon_sha256":"e10e4f63334faef26b691680f853a85de317034883f61c563937b850365a8376","abstract_canon_sha256":"dbfa9b75f3c347929c0dd2c6c0afcec3234f5e52095a1d754eaed9000911077e"},"schema_version":"1.0"},"canonical_sha256":"97f14988cb8ad5ded5cb6e02c4d9100c9a961ad4c02912254c82632a94c56fab","source":{"kind":"arxiv","id":"1603.01296","version":3},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.01296","created_at":"2026-05-18T00:22:29Z"},{"alias_kind":"arxiv_version","alias_value":"1603.01296v3","created_at":"2026-05-18T00:22:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.01296","created_at":"2026-05-18T00:22:29Z"},{"alias_kind":"pith_short_12","alias_value":"S7YUTCGLRLK5","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_16","alias_value":"S7YUTCGLRLK55VOL","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_8","alias_value":"S7YUTCGL","created_at":"2026-05-18T12:30:44Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:S7YUTCGLRLK55VOLNYBMJWIQBS","target":"record","payload":{"canonical_record":{"source":{"id":"1603.01296","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-03-03T21:59:05Z","cross_cats_sorted":[],"title_canon_sha256":"e10e4f63334faef26b691680f853a85de317034883f61c563937b850365a8376","abstract_canon_sha256":"dbfa9b75f3c347929c0dd2c6c0afcec3234f5e52095a1d754eaed9000911077e"},"schema_version":"1.0"},"canonical_sha256":"97f14988cb8ad5ded5cb6e02c4d9100c9a961ad4c02912254c82632a94c56fab","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:22:29.891647Z","signature_b64":"6+AVj+dtDQ+KAb1DW6M6vwB8055+/rtNHMaG8AqVdDJc5cdmFoJSQWxLsxfaYXLI7TNmpfLpwOdHDmZg69A9AA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"97f14988cb8ad5ded5cb6e02c4d9100c9a961ad4c02912254c82632a94c56fab","last_reissued_at":"2026-05-18T00:22:29.890992Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:22:29.890992Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1603.01296","source_version":3,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:22:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"1NQtSX21GmaLcKsNEEbGWOZA2YHcXIhl9wQvmKanjhJLa2F/4hjugrlwXAKGnjlgqwcRgeOaURmZ+QiJ8p+VCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T15:28:36.652762Z"},"content_sha256":"f6b65c272407a1e2c9be207b3eaa8db7d1526193bff3346ea87f4400baa24bee","schema_version":"1.0","event_id":"sha256:f6b65c272407a1e2c9be207b3eaa8db7d1526193bff3346ea87f4400baa24bee"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:S7YUTCGLRLK55VOLNYBMJWIQBS","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On the class numbers of the fields of the $p^n$-torsion points of elliptic curves over $\\mathbb{Q}$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Fumio Sairaiji, Takuya Yamauchi","submitted_at":"2016-03-03T21:59:05Z","abstract_excerpt":"Let $E$ be an elliptic curve over $\\mathbb{Q}$ which has multiplicative reduction at a fixed prime $p$. For each positive integer $n$ we put $K_n:=\\mathbb{Q}(E[p^n])$. The aim of this paper is to extend the author's previous our results concerning with the order of the $p$-Sylow group of the ideal class group of $K_n$ to more general setting. We also modify the previous lower bound of the order and describe the new lower bound in terms of the Mordell-Weil rank of $E(\\mathbb{Q})$ and the ramification related to $E$."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.01296","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:22:29Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"4H/qKzVpULfwJ5kZ8WrpTTqzFzu9BgYlW+AcsFgFoIHfyYyLOzWRg7Vr0PYaUisuxLdpfD4Aylg9pL8xuj6vAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-01T15:28:36.653102Z"},"content_sha256":"e55ab1fea7900b04c062fa77f63f96ffafe50cfcb73ccc67831ba06acc5cbee4","schema_version":"1.0","event_id":"sha256:e55ab1fea7900b04c062fa77f63f96ffafe50cfcb73ccc67831ba06acc5cbee4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/S7YUTCGLRLK55VOLNYBMJWIQBS/bundle.json","state_url":"https://pith.science/pith/S7YUTCGLRLK55VOLNYBMJWIQBS/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/S7YUTCGLRLK55VOLNYBMJWIQBS/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-01T15:28:36Z","links":{"resolver":"https://pith.science/pith/S7YUTCGLRLK55VOLNYBMJWIQBS","bundle":"https://pith.science/pith/S7YUTCGLRLK55VOLNYBMJWIQBS/bundle.json","state":"https://pith.science/pith/S7YUTCGLRLK55VOLNYBMJWIQBS/state.json","well_known_bundle":"https://pith.science/.well-known/pith/S7YUTCGLRLK55VOLNYBMJWIQBS/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:S7YUTCGLRLK55VOLNYBMJWIQBS","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":"dbfa9b75f3c347929c0dd2c6c0afcec3234f5e52095a1d754eaed9000911077e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-03-03T21:59:05Z","title_canon_sha256":"e10e4f63334faef26b691680f853a85de317034883f61c563937b850365a8376"},"schema_version":"1.0","source":{"id":"1603.01296","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.01296","created_at":"2026-05-18T00:22:29Z"},{"alias_kind":"arxiv_version","alias_value":"1603.01296v3","created_at":"2026-05-18T00:22:29Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.01296","created_at":"2026-05-18T00:22:29Z"},{"alias_kind":"pith_short_12","alias_value":"S7YUTCGLRLK5","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_16","alias_value":"S7YUTCGLRLK55VOL","created_at":"2026-05-18T12:30:44Z"},{"alias_kind":"pith_short_8","alias_value":"S7YUTCGL","created_at":"2026-05-18T12:30:44Z"}],"graph_snapshots":[{"event_id":"sha256:e55ab1fea7900b04c062fa77f63f96ffafe50cfcb73ccc67831ba06acc5cbee4","target":"graph","created_at":"2026-05-18T00:22:29Z","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":"Let $E$ be an elliptic curve over $\\mathbb{Q}$ which has multiplicative reduction at a fixed prime $p$. For each positive integer $n$ we put $K_n:=\\mathbb{Q}(E[p^n])$. The aim of this paper is to extend the author's previous our results concerning with the order of the $p$-Sylow group of the ideal class group of $K_n$ to more general setting. We also modify the previous lower bound of the order and describe the new lower bound in terms of the Mordell-Weil rank of $E(\\mathbb{Q})$ and the ramification related to $E$.","authors_text":"Fumio Sairaiji, Takuya Yamauchi","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-03-03T21:59:05Z","title":"On the class numbers of the fields of the $p^n$-torsion points of elliptic curves over $\\mathbb{Q}$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.01296","kind":"arxiv","version":3},"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:f6b65c272407a1e2c9be207b3eaa8db7d1526193bff3346ea87f4400baa24bee","target":"record","created_at":"2026-05-18T00:22:29Z","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":"dbfa9b75f3c347929c0dd2c6c0afcec3234f5e52095a1d754eaed9000911077e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-03-03T21:59:05Z","title_canon_sha256":"e10e4f63334faef26b691680f853a85de317034883f61c563937b850365a8376"},"schema_version":"1.0","source":{"id":"1603.01296","kind":"arxiv","version":3}},"canonical_sha256":"97f14988cb8ad5ded5cb6e02c4d9100c9a961ad4c02912254c82632a94c56fab","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"97f14988cb8ad5ded5cb6e02c4d9100c9a961ad4c02912254c82632a94c56fab","first_computed_at":"2026-05-18T00:22:29.890992Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:22:29.890992Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"6+AVj+dtDQ+KAb1DW6M6vwB8055+/rtNHMaG8AqVdDJc5cdmFoJSQWxLsxfaYXLI7TNmpfLpwOdHDmZg69A9AA==","signature_status":"signed_v1","signed_at":"2026-05-18T00:22:29.891647Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.01296","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f6b65c272407a1e2c9be207b3eaa8db7d1526193bff3346ea87f4400baa24bee","sha256:e55ab1fea7900b04c062fa77f63f96ffafe50cfcb73ccc67831ba06acc5cbee4"],"state_sha256":"14d4641b7e0d857c186ebb47a8ee4cc26d8328dbc33f3d074dbe0cb69f59d972"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"Hgm9wNaZoJRt20pjHBiNLZHQpQeF/wisELnyEpYV7OcFao4Gr6w0X2lc9I5bUFIa77P6UFZZiAY0NknOqD8ODA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-01T15:28:36.655005Z","bundle_sha256":"bf0e2b5934263aee769bc3b73e6cf2f0d218c4bfa41e84570838a265d67bb06e"}}