{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2021:N6YVHBEJCYY7EZHNTF2PMR3HJC","short_pith_number":"pith:N6YVHBEJ","canonical_record":{"source":{"id":"2110.09063","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2021-10-18T07:11:58Z","cross_cats_sorted":[],"title_canon_sha256":"7bcebb600928eaeac596c55e5fdffba7400c3433db75c924a7d052b127735df6","abstract_canon_sha256":"7cbd26479cd065d4dfc7bcc8e77e2d47e30a54d745e17b009ad4c418fb4916d7"},"schema_version":"1.0"},"canonical_sha256":"6fb15384891631f264ed9974f64767488f70e4fa849ac13c78532a7f59f05852","source":{"kind":"arxiv","id":"2110.09063","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2110.09063","created_at":"2026-07-05T03:23:23Z"},{"alias_kind":"arxiv_version","alias_value":"2110.09063v1","created_at":"2026-07-05T03:23:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2110.09063","created_at":"2026-07-05T03:23:23Z"},{"alias_kind":"pith_short_12","alias_value":"N6YVHBEJCYY7","created_at":"2026-07-05T03:23:23Z"},{"alias_kind":"pith_short_16","alias_value":"N6YVHBEJCYY7EZHN","created_at":"2026-07-05T03:23:23Z"},{"alias_kind":"pith_short_8","alias_value":"N6YVHBEJ","created_at":"2026-07-05T03:23:23Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2021:N6YVHBEJCYY7EZHNTF2PMR3HJC","target":"record","payload":{"canonical_record":{"source":{"id":"2110.09063","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2021-10-18T07:11:58Z","cross_cats_sorted":[],"title_canon_sha256":"7bcebb600928eaeac596c55e5fdffba7400c3433db75c924a7d052b127735df6","abstract_canon_sha256":"7cbd26479cd065d4dfc7bcc8e77e2d47e30a54d745e17b009ad4c418fb4916d7"},"schema_version":"1.0"},"canonical_sha256":"6fb15384891631f264ed9974f64767488f70e4fa849ac13c78532a7f59f05852","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:23:23.716669Z","signature_b64":"NRvaZNPROAMxA7ZSGqpqyBDBa1/ssALytuuzVhnF7Es1znwm1Wc5H4C+Hf5N1AjUi2e95oUTFZpAdGYLAa3tCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"6fb15384891631f264ed9974f64767488f70e4fa849ac13c78532a7f59f05852","last_reissued_at":"2026-07-05T03:23:23.716246Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:23:23.716246Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2110.09063","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:23:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"EkGE7jhely2mef0QZCTAJD2BWrBgYS77mzWaHDCm0E6DXgyFesgbrYt0oSrMoH17G6Zt7Riv3Omz5iaoroPnCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T07:17:35.157189Z"},"content_sha256":"ca45a08890ef3f796a1b3320427818928989e2ebe55d354aedc39d827b544dcb","schema_version":"1.0","event_id":"sha256:ca45a08890ef3f796a1b3320427818928989e2ebe55d354aedc39d827b544dcb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2021:N6YVHBEJCYY7EZHNTF2PMR3HJC","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The second moment of the size of the $2$-Selmer group of elliptic curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Arul Shankar, Ashvin Swaminathan, Manjul Bhargava","submitted_at":"2021-10-18T07:11:58Z","abstract_excerpt":"In this paper, we prove that when elliptic curves over $\\mathbb{Q}$ are ordered by height, the second moment of the size of the $2$-Selmer group is at most $15$. This confirms a conjecture of Poonen and Rains."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2110.09063","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/2110.09063/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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-07-05T03:23:23Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"63HaXHNYYeSGmS7AHReEjv1kr4WmMuCGc8h6nB7lvCTN1TIebgQQjyLGZW+h3vQu3PgeytuIpFO6kGk3s3VIAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-06T07:17:35.158271Z"},"content_sha256":"c0c7e660baf71dd37e7cb958c6276ef0436fc9fc9548f2df0fcef32a8e27cd04","schema_version":"1.0","event_id":"sha256:c0c7e660baf71dd37e7cb958c6276ef0436fc9fc9548f2df0fcef32a8e27cd04"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/N6YVHBEJCYY7EZHNTF2PMR3HJC/bundle.json","state_url":"https://pith.science/pith/N6YVHBEJCYY7EZHNTF2PMR3HJC/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/N6YVHBEJCYY7EZHNTF2PMR3HJC/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":"2026-08-06T07:17:35Z","links":{"resolver":"https://pith.science/pith/N6YVHBEJCYY7EZHNTF2PMR3HJC","bundle":"https://pith.science/pith/N6YVHBEJCYY7EZHNTF2PMR3HJC/bundle.json","state":"https://pith.science/pith/N6YVHBEJCYY7EZHNTF2PMR3HJC/state.json","well_known_bundle":"https://pith.science/.well-known/pith/N6YVHBEJCYY7EZHNTF2PMR3HJC/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2021:N6YVHBEJCYY7EZHNTF2PMR3HJC","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":"7cbd26479cd065d4dfc7bcc8e77e2d47e30a54d745e17b009ad4c418fb4916d7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2021-10-18T07:11:58Z","title_canon_sha256":"7bcebb600928eaeac596c55e5fdffba7400c3433db75c924a7d052b127735df6"},"schema_version":"1.0","source":{"id":"2110.09063","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2110.09063","created_at":"2026-07-05T03:23:23Z"},{"alias_kind":"arxiv_version","alias_value":"2110.09063v1","created_at":"2026-07-05T03:23:23Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2110.09063","created_at":"2026-07-05T03:23:23Z"},{"alias_kind":"pith_short_12","alias_value":"N6YVHBEJCYY7","created_at":"2026-07-05T03:23:23Z"},{"alias_kind":"pith_short_16","alias_value":"N6YVHBEJCYY7EZHN","created_at":"2026-07-05T03:23:23Z"},{"alias_kind":"pith_short_8","alias_value":"N6YVHBEJ","created_at":"2026-07-05T03:23:23Z"}],"graph_snapshots":[{"event_id":"sha256:c0c7e660baf71dd37e7cb958c6276ef0436fc9fc9548f2df0fcef32a8e27cd04","target":"graph","created_at":"2026-07-05T03:23:23Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2110.09063/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"In this paper, we prove that when elliptic curves over $\\mathbb{Q}$ are ordered by height, the second moment of the size of the $2$-Selmer group is at most $15$. This confirms a conjecture of Poonen and Rains.","authors_text":"Arul Shankar, Ashvin Swaminathan, Manjul Bhargava","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2021-10-18T07:11:58Z","title":"The second moment of the size of the $2$-Selmer group of elliptic curves"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2110.09063","kind":"arxiv","version":1},"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:ca45a08890ef3f796a1b3320427818928989e2ebe55d354aedc39d827b544dcb","target":"record","created_at":"2026-07-05T03:23:23Z","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":"7cbd26479cd065d4dfc7bcc8e77e2d47e30a54d745e17b009ad4c418fb4916d7","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2021-10-18T07:11:58Z","title_canon_sha256":"7bcebb600928eaeac596c55e5fdffba7400c3433db75c924a7d052b127735df6"},"schema_version":"1.0","source":{"id":"2110.09063","kind":"arxiv","version":1}},"canonical_sha256":"6fb15384891631f264ed9974f64767488f70e4fa849ac13c78532a7f59f05852","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"6fb15384891631f264ed9974f64767488f70e4fa849ac13c78532a7f59f05852","first_computed_at":"2026-07-05T03:23:23.716246Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:23:23.716246Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"NRvaZNPROAMxA7ZSGqpqyBDBa1/ssALytuuzVhnF7Es1znwm1Wc5H4C+Hf5N1AjUi2e95oUTFZpAdGYLAa3tCA==","signature_status":"signed_v1","signed_at":"2026-07-05T03:23:23.716669Z","signed_message":"canonical_sha256_bytes"},"source_id":"2110.09063","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:ca45a08890ef3f796a1b3320427818928989e2ebe55d354aedc39d827b544dcb","sha256:c0c7e660baf71dd37e7cb958c6276ef0436fc9fc9548f2df0fcef32a8e27cd04"],"state_sha256":"dd65d3552f384091321474a646b66cdab25e6b656519e2c816626e362c00c405"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"C9b+q7Wd56mjRoLyT0d/9jYnzhtu2Zb7ji0PGrkbn6gSs3/5MGHPFBZBIei4R8TbLvV7N3PX5wzV4lobu4PwCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-06T07:17:35.165962Z","bundle_sha256":"e39d0f349ae26fd40bcccd0e85d3bba6dabbba947e8ee2ea6bae340a63e2c43f"}}