{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2024:QGWBBJTJ3YUHZEDJKIT3ZKOOQR","short_pith_number":"pith:QGWBBJTJ","canonical_record":{"source":{"id":"2412.00995","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-12-01T22:56:00Z","cross_cats_sorted":[],"title_canon_sha256":"68ce4f51506b714b545ecefe29014315092d93fc742f88405f9c33926d8053f6","abstract_canon_sha256":"0ebef8de736d8b0d33c9cb7aa57139a6e32283eb71026c7682b77c4299478af2"},"schema_version":"1.0"},"canonical_sha256":"81ac10a669de287c90695227bca9ce84695da9eb861794e8ad14419b356f3711","source":{"kind":"arxiv","id":"2412.00995","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.00995","created_at":"2026-07-05T09:42:51Z"},{"alias_kind":"arxiv_version","alias_value":"2412.00995v1","created_at":"2026-07-05T09:42:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.00995","created_at":"2026-07-05T09:42:51Z"},{"alias_kind":"pith_short_12","alias_value":"QGWBBJTJ3YUH","created_at":"2026-07-05T09:42:51Z"},{"alias_kind":"pith_short_16","alias_value":"QGWBBJTJ3YUHZEDJ","created_at":"2026-07-05T09:42:51Z"},{"alias_kind":"pith_short_8","alias_value":"QGWBBJTJ","created_at":"2026-07-05T09:42:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2024:QGWBBJTJ3YUHZEDJKIT3ZKOOQR","target":"record","payload":{"canonical_record":{"source":{"id":"2412.00995","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-12-01T22:56:00Z","cross_cats_sorted":[],"title_canon_sha256":"68ce4f51506b714b545ecefe29014315092d93fc742f88405f9c33926d8053f6","abstract_canon_sha256":"0ebef8de736d8b0d33c9cb7aa57139a6e32283eb71026c7682b77c4299478af2"},"schema_version":"1.0"},"canonical_sha256":"81ac10a669de287c90695227bca9ce84695da9eb861794e8ad14419b356f3711","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:42:51.723747Z","signature_b64":"msu2m1kJZdMIdem/uaQvnqGIBt10/GX7231nWw99FOmkCZoVhlX0oZ8uEzwKjSKrPHl7y25ecJqyCoc/BMizCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"81ac10a669de287c90695227bca9ce84695da9eb861794e8ad14419b356f3711","last_reissued_at":"2026-07-05T09:42:51.722941Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:42:51.722941Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2412.00995","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":"2026-07-05T09:42:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"mFIwTvFLrrysvMtDlYa11Z4tuvlZB18FHZI8p7MafZ27ZPFRTlYHor8tt/YYH4UtFPGKUDtCVhA5Ri1knp6DDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T20:52:41.981148Z"},"content_sha256":"d95ba85948f2764ec76fec6b5a68668142cde6dd1dd63718563ae2251929a3d5","schema_version":"1.0","event_id":"sha256:d95ba85948f2764ec76fec6b5a68668142cde6dd1dd63718563ae2251929a3d5"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2024:QGWBBJTJ3YUHZEDJKIT3ZKOOQR","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Secondary terms in the first moment of $|{\\rm Sel}_2(E)|$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Arul Shankar, Takashi Taniguchi","submitted_at":"2024-12-01T22:56:00Z","abstract_excerpt":"We prove the existence of secondary terms of order $X^{3/4}$, with power saving error terms, in the counting functions of $|{\\rm Sel}_2(E)|$, the 2-Selmer group of E, for elliptic curves E having height bounded by X. This is the first improvement on the error term of $o(X^{5/6})$, proved by Bhargava--Shankar, where the primary term of order $X^{5/6}$ for this counting function was obtained."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.00995","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/2412.00995/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":"2026-07-05T09:42:51Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"KLHutsd/1g47KEOJUD7Nrt10EmlByv+BW59UN63KNMDOSwnfct/okY98pf2I+DoyQEYkMc0Hk0GrQlWvk/sBAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-17T20:52:41.983021Z"},"content_sha256":"1a2d6bd9015e4295ac084618f2380d86be72f8ef13468d0eee10296a61af7632","schema_version":"1.0","event_id":"sha256:1a2d6bd9015e4295ac084618f2380d86be72f8ef13468d0eee10296a61af7632"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/QGWBBJTJ3YUHZEDJKIT3ZKOOQR/bundle.json","state_url":"https://pith.science/pith/QGWBBJTJ3YUHZEDJKIT3ZKOOQR/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/QGWBBJTJ3YUHZEDJKIT3ZKOOQR/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":"2026-08-17T20:52:41Z","links":{"resolver":"https://pith.science/pith/QGWBBJTJ3YUHZEDJKIT3ZKOOQR","bundle":"https://pith.science/pith/QGWBBJTJ3YUHZEDJKIT3ZKOOQR/bundle.json","state":"https://pith.science/pith/QGWBBJTJ3YUHZEDJKIT3ZKOOQR/state.json","well_known_bundle":"https://pith.science/.well-known/pith/QGWBBJTJ3YUHZEDJKIT3ZKOOQR/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:QGWBBJTJ3YUHZEDJKIT3ZKOOQR","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":"0ebef8de736d8b0d33c9cb7aa57139a6e32283eb71026c7682b77c4299478af2","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-12-01T22:56:00Z","title_canon_sha256":"68ce4f51506b714b545ecefe29014315092d93fc742f88405f9c33926d8053f6"},"schema_version":"1.0","source":{"id":"2412.00995","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.00995","created_at":"2026-07-05T09:42:51Z"},{"alias_kind":"arxiv_version","alias_value":"2412.00995v1","created_at":"2026-07-05T09:42:51Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.00995","created_at":"2026-07-05T09:42:51Z"},{"alias_kind":"pith_short_12","alias_value":"QGWBBJTJ3YUH","created_at":"2026-07-05T09:42:51Z"},{"alias_kind":"pith_short_16","alias_value":"QGWBBJTJ3YUHZEDJ","created_at":"2026-07-05T09:42:51Z"},{"alias_kind":"pith_short_8","alias_value":"QGWBBJTJ","created_at":"2026-07-05T09:42:51Z"}],"graph_snapshots":[{"event_id":"sha256:1a2d6bd9015e4295ac084618f2380d86be72f8ef13468d0eee10296a61af7632","target":"graph","created_at":"2026-07-05T09:42:51Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2412.00995/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We prove the existence of secondary terms of order $X^{3/4}$, with power saving error terms, in the counting functions of $|{\\rm Sel}_2(E)|$, the 2-Selmer group of E, for elliptic curves E having height bounded by X. This is the first improvement on the error term of $o(X^{5/6})$, proved by Bhargava--Shankar, where the primary term of order $X^{5/6}$ for this counting function was obtained.","authors_text":"Arul Shankar, Takashi Taniguchi","cross_cats":[],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-12-01T22:56:00Z","title":"Secondary terms in the first moment of $|{\\rm Sel}_2(E)|$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.00995","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:d95ba85948f2764ec76fec6b5a68668142cde6dd1dd63718563ae2251929a3d5","target":"record","created_at":"2026-07-05T09:42:51Z","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":"0ebef8de736d8b0d33c9cb7aa57139a6e32283eb71026c7682b77c4299478af2","cross_cats_sorted":[],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2024-12-01T22:56:00Z","title_canon_sha256":"68ce4f51506b714b545ecefe29014315092d93fc742f88405f9c33926d8053f6"},"schema_version":"1.0","source":{"id":"2412.00995","kind":"arxiv","version":1}},"canonical_sha256":"81ac10a669de287c90695227bca9ce84695da9eb861794e8ad14419b356f3711","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"81ac10a669de287c90695227bca9ce84695da9eb861794e8ad14419b356f3711","first_computed_at":"2026-07-05T09:42:51.722941Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T09:42:51.722941Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"msu2m1kJZdMIdem/uaQvnqGIBt10/GX7231nWw99FOmkCZoVhlX0oZ8uEzwKjSKrPHl7y25ecJqyCoc/BMizCQ==","signature_status":"signed_v1","signed_at":"2026-07-05T09:42:51.723747Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.00995","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:d95ba85948f2764ec76fec6b5a68668142cde6dd1dd63718563ae2251929a3d5","sha256:1a2d6bd9015e4295ac084618f2380d86be72f8ef13468d0eee10296a61af7632"],"state_sha256":"ab716b9cb6f89e04dc0852eca0d9e9acd89fba99b96f977787c6af38076ec902"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6nQVPLgcprjOpGgJth0VyAKL9CaPpqbfXVBpSTq4bZ9olWWfCYGVf6zOEoiuXTxAXE/qyXNEIvIhnmdX3yzFBA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-17T20:52:41.993179Z","bundle_sha256":"8cce31dcb8fab2b2f328f28250fd72dd089b5f122b91122e440829d0d247ffb9"}}