{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2021:T42HOXA5IMXZT564TCVRJNEAJD","short_pith_number":"pith:T42HOXA5","schema_version":"1.0","canonical_sha256":"9f34775c1d432f99f7dc98ab14b48048cc7d8246ff6285cbd2dddebd4c5d492c","source":{"kind":"arxiv","id":"2108.10401","version":1},"attestation_state":"computed","paper":{"title":"Quadratic forms in 8 prime variables","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.NT","authors_text":"Ben Green","submitted_at":"2021-08-23T20:49:55Z","abstract_excerpt":"We give an asymptotic for the number of prime solutions to $Q(x_1,\\dots, x_8) = N$, subject to a mild non-degeneracy condition on the homogeneous quadratic form $Q$.\n  The argument initially proceeds via the circle method, but this does not suffice by itself. To obtain a nontrivial bound on certain averages of exponential sums, we interpret these sums as matrix coefficients for the Weil representation of the symplectic group $\\operatorname{Sp}_8(\\mathbf{Z}/q\\mathbf{Z})$. Averages of such matrix coefficients are then bounded using an amplification argument and a convergence result for convoluti"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2108.10401","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2021-08-23T20:49:55Z","cross_cats_sorted":["math.RT"],"title_canon_sha256":"2a9efb1ca5d3737c80d616f0a0151babed8f81566a7c0df1fa8f41d0db99d0e9","abstract_canon_sha256":"37affda8c9dddd4cfed4e69ea06bddf2a0665c20eac161b42a42ae0fcefd781a"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:08:09.433601Z","signature_b64":"KD5GUNr/0qIFqA3GLxa/q53zws9mI2Sr8vVP9+Ee5qwJlEpcCb+x4dOb5gbyjyBxSuRR+nAao/t0LF8kQ4KPCw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9f34775c1d432f99f7dc98ab14b48048cc7d8246ff6285cbd2dddebd4c5d492c","last_reissued_at":"2026-07-05T03:08:09.433194Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:08:09.433194Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Quadratic forms in 8 prime variables","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.RT"],"primary_cat":"math.NT","authors_text":"Ben Green","submitted_at":"2021-08-23T20:49:55Z","abstract_excerpt":"We give an asymptotic for the number of prime solutions to $Q(x_1,\\dots, x_8) = N$, subject to a mild non-degeneracy condition on the homogeneous quadratic form $Q$.\n  The argument initially proceeds via the circle method, but this does not suffice by itself. To obtain a nontrivial bound on certain averages of exponential sums, we interpret these sums as matrix coefficients for the Weil representation of the symplectic group $\\operatorname{Sp}_8(\\mathbf{Z}/q\\mathbf{Z})$. Averages of such matrix coefficients are then bounded using an amplification argument and a convergence result for convoluti"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2108.10401","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/2108.10401/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2108.10401","created_at":"2026-07-05T03:08:09.433264+00:00"},{"alias_kind":"arxiv_version","alias_value":"2108.10401v1","created_at":"2026-07-05T03:08:09.433264+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2108.10401","created_at":"2026-07-05T03:08:09.433264+00:00"},{"alias_kind":"pith_short_12","alias_value":"T42HOXA5IMXZ","created_at":"2026-07-05T03:08:09.433264+00:00"},{"alias_kind":"pith_short_16","alias_value":"T42HOXA5IMXZT564","created_at":"2026-07-05T03:08:09.433264+00:00"},{"alias_kind":"pith_short_8","alias_value":"T42HOXA5","created_at":"2026-07-05T03:08:09.433264+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":1,"internal_anchor_count":1,"sample":[{"citing_arxiv_id":"2501.15033","citing_title":"Anisotropic quadratic equations in three variables","ref_index":10,"is_internal_anchor":true}]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/T42HOXA5IMXZT564TCVRJNEAJD","json":"https://pith.science/pith/T42HOXA5IMXZT564TCVRJNEAJD.json","graph_json":"https://pith.science/api/pith-number/T42HOXA5IMXZT564TCVRJNEAJD/graph.json","events_json":"https://pith.science/api/pith-number/T42HOXA5IMXZT564TCVRJNEAJD/events.json","paper":"https://pith.science/paper/T42HOXA5"},"agent_actions":{"view_html":"https://pith.science/pith/T42HOXA5IMXZT564TCVRJNEAJD","download_json":"https://pith.science/pith/T42HOXA5IMXZT564TCVRJNEAJD.json","view_paper":"https://pith.science/paper/T42HOXA5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2108.10401&json=true","fetch_graph":"https://pith.science/api/pith-number/T42HOXA5IMXZT564TCVRJNEAJD/graph.json","fetch_events":"https://pith.science/api/pith-number/T42HOXA5IMXZT564TCVRJNEAJD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/T42HOXA5IMXZT564TCVRJNEAJD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/T42HOXA5IMXZT564TCVRJNEAJD/action/storage_attestation","attest_author":"https://pith.science/pith/T42HOXA5IMXZT564TCVRJNEAJD/action/author_attestation","sign_citation":"https://pith.science/pith/T42HOXA5IMXZT564TCVRJNEAJD/action/citation_signature","submit_replication":"https://pith.science/pith/T42HOXA5IMXZT564TCVRJNEAJD/action/replication_record"}},"created_at":"2026-07-05T03:08:09.433264+00:00","updated_at":"2026-07-05T03:08:09.433264+00:00"}