{"paper":{"title":"Degree three cohomology of function fields of surfaces","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"R. Parimala, V. Suresh","submitted_at":"2010-12-24T09:12:17Z","abstract_excerpt":"Let F be a finite field and l a prime not equal to the characteristic of F. Let K be the function field of a surface over F. Assume that K contains a primitive lth root of unity. In the paper we prove a certain local-global principle for elements of H^3(K, {\\mu}_l) in terms of symbols in H^2(K, {\\mu}_l) with respect to discrete valuations of K. We also show that this local global principle is equivalent to the vanishing of certain unramified cohomology groups of 3-folds over finite fields. Using this local-global principle we show that every element in H^3(F, {\\mu}_l) is a symbol. The vanishin"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1012.5367","kind":"arxiv","version":2},"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"}