{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2022:UYNUOWSXH6U3W73TGPVL5VZWEZ","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":"ea88a2fc9444b7a4d71e646cf7988631f08882a32163d07665489fd7bfc49926","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-07-07T14:14:02Z","title_canon_sha256":"4f51c785e51a639c93d9139d5e2cd7650b366338bd4b78e955aa7ff9cd380dd8"},"schema_version":"1.0","source":{"id":"2207.03309","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2207.03309","created_at":"2026-07-05T04:40:28Z"},{"alias_kind":"arxiv_version","alias_value":"2207.03309v2","created_at":"2026-07-05T04:40:28Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2207.03309","created_at":"2026-07-05T04:40:28Z"},{"alias_kind":"pith_short_12","alias_value":"UYNUOWSXH6U3","created_at":"2026-07-05T04:40:28Z"},{"alias_kind":"pith_short_16","alias_value":"UYNUOWSXH6U3W73T","created_at":"2026-07-05T04:40:28Z"},{"alias_kind":"pith_short_8","alias_value":"UYNUOWSX","created_at":"2026-07-05T04:40:28Z"}],"graph_snapshots":[{"event_id":"sha256:273bf3130e4755d184771bacfba6a3148b84bc456e13aa6e451fca3d0c45a3c9","target":"graph","created_at":"2026-07-05T04:40:28Z","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/2207.03309/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"We determine average sizes/bounds for the $2$- and $3$-Selmer groups in various families of elliptic curves with marked points, thus confirming several cases of the Poonen--Rains heuristics. As a consequence, we deduce that the average ranks of the elliptic curves in all of these families are bounded. Our proofs are uniform and make use of parametrizations involving various forms of $2 \\times 2 \\times 2 \\times 2$ and $3 \\times 3 \\times 3$ matrices that we studied in a previous paper.\n  We also deduce that $100\\%$ of genus one curves of the form $y^2 = Ax^4 + Bx^2 z^2 + Cz^4$ with $A, B, C \\in ","authors_text":"Manjul Bhargava, Wei Ho","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-07-07T14:14:02Z","title":"On average sizes of Selmer groups and ranks in families of elliptic curves having marked points"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2207.03309","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":"sha256:211472c74b98326432477614f68845ff46e406ca8349ea7fecb4f30e038316f0","target":"record","created_at":"2026-07-05T04:40:28Z","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":"ea88a2fc9444b7a4d71e646cf7988631f08882a32163d07665489fd7bfc49926","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2022-07-07T14:14:02Z","title_canon_sha256":"4f51c785e51a639c93d9139d5e2cd7650b366338bd4b78e955aa7ff9cd380dd8"},"schema_version":"1.0","source":{"id":"2207.03309","kind":"arxiv","version":2}},"canonical_sha256":"a61b475a573fa9bb7f7333eabed7362668ebf2b7bd2ac74cac976f7fe93a08ba","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"a61b475a573fa9bb7f7333eabed7362668ebf2b7bd2ac74cac976f7fe93a08ba","first_computed_at":"2026-07-05T04:40:28.221722Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T04:40:28.221722Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"yhJZM9+D3xwjSRtADfGzkEacTO3+4rWb946iT8GqlSFsGGGh72VspLkpugTPEg6Z14gDwitMAeI2YJCIxLmWBA==","signature_status":"signed_v1","signed_at":"2026-07-05T04:40:28.222235Z","signed_message":"canonical_sha256_bytes"},"source_id":"2207.03309","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:211472c74b98326432477614f68845ff46e406ca8349ea7fecb4f30e038316f0","sha256:273bf3130e4755d184771bacfba6a3148b84bc456e13aa6e451fca3d0c45a3c9"],"state_sha256":"24f09225c6ffdc05e745047cd238c1ed8cee8584e53fcade6a6e41b77ce4f6e1"}