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Voskresenski, ed.), Kuibyshev State University, Kuibyshev, 1989, pp. 57--67.\n  Let $X$ be be an absolutely irreducible smooth projective surface over a finite field $k$ of odd characteristic, let $Br(X)$ be the (commutative periodic) Brauer group of $X$ and $DIV Br(X)$ the subgroup of its divisible elements. We write $Br(X)_{DIV}$ for the quotient $Br(X)/DIV Br(X)$ and $Br(X)_{DIV}(2)$ for its (finite) $2$-primary component. We prove that the order of $Br("},"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":"1802.01776","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-02-06T03:13:58Z","cross_cats_sorted":[],"title_canon_sha256":"978900915d5454d13575b139dd796201b38c07c354b0460c51cc2b7e2b22e2a9","abstract_canon_sha256":"742bd11f60c14ea8980bff457ac7061e536f1c9be83385189dd23c1443c8eb34"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:24:21.332098Z","signature_b64":"hIpaJxIXojhTtSFMIFBG4e5PidlWoyebyFxPQveVPO8QvkNDFzdbzwPm8YkLRNVA69SWfiqpSjgAgcaQcFO9DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"087f55b60908fa7f9008a4dbb2197e17a67ba5d7e0002b6e310a0ac1cb61a70f","last_reissued_at":"2026-05-18T00:24:21.331540Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:24:21.331540Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"The Brauer Group of a Surface over a Finite Field","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Yuri G. Zarhin","submitted_at":"2018-02-06T03:13:58Z","abstract_excerpt":"This is an English translation of the author's 1989 note in Russian, published in a collection \"Arithmetic and Geometry of Varieties\" (V.E. Voskresenski, ed.), Kuibyshev State University, Kuibyshev, 1989, pp. 57--67.\n  Let $X$ be be an absolutely irreducible smooth projective surface over a finite field $k$ of odd characteristic, let $Br(X)$ be the (commutative periodic) Brauer group of $X$ and $DIV Br(X)$ the subgroup of its divisible elements. We write $Br(X)_{DIV}$ for the quotient $Br(X)/DIV Br(X)$ and $Br(X)_{DIV}(2)$ for its (finite) $2$-primary component. We prove that the order of $Br("},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1802.01776","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":""},"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":"1802.01776","created_at":"2026-05-18T00:24:21.331612+00:00"},{"alias_kind":"arxiv_version","alias_value":"1802.01776v1","created_at":"2026-05-18T00:24:21.331612+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1802.01776","created_at":"2026-05-18T00:24:21.331612+00:00"},{"alias_kind":"pith_short_12","alias_value":"BB7VLNQJBD5H","created_at":"2026-05-18T12:32:13.499390+00:00"},{"alias_kind":"pith_short_16","alias_value":"BB7VLNQJBD5H7EAI","created_at":"2026-05-18T12:32:13.499390+00:00"},{"alias_kind":"pith_short_8","alias_value":"BB7VLNQJ","created_at":"2026-05-18T12:32:13.499390+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/BB7VLNQJBD5H7EAIUTN3EGL6C6","json":"https://pith.science/pith/BB7VLNQJBD5H7EAIUTN3EGL6C6.json","graph_json":"https://pith.science/api/pith-number/BB7VLNQJBD5H7EAIUTN3EGL6C6/graph.json","events_json":"https://pith.science/api/pith-number/BB7VLNQJBD5H7EAIUTN3EGL6C6/events.json","paper":"https://pith.science/paper/BB7VLNQJ"},"agent_actions":{"view_html":"https://pith.science/pith/BB7VLNQJBD5H7EAIUTN3EGL6C6","download_json":"https://pith.science/pith/BB7VLNQJBD5H7EAIUTN3EGL6C6.json","view_paper":"https://pith.science/paper/BB7VLNQJ","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1802.01776&json=true","fetch_graph":"https://pith.science/api/pith-number/BB7VLNQJBD5H7EAIUTN3EGL6C6/graph.json","fetch_events":"https://pith.science/api/pith-number/BB7VLNQJBD5H7EAIUTN3EGL6C6/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/BB7VLNQJBD5H7EAIUTN3EGL6C6/action/timestamp_anchor","attest_storage":"https://pith.science/pith/BB7VLNQJBD5H7EAIUTN3EGL6C6/action/storage_attestation","attest_author":"https://pith.science/pith/BB7VLNQJBD5H7EAIUTN3EGL6C6/action/author_attestation","sign_citation":"https://pith.science/pith/BB7VLNQJBD5H7EAIUTN3EGL6C6/action/citation_signature","submit_replication":"https://pith.science/pith/BB7VLNQJBD5H7EAIUTN3EGL6C6/action/replication_record"}},"created_at":"2026-05-18T00:24:21.331612+00:00","updated_at":"2026-05-18T00:24:21.331612+00:00"}