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This map is a generalization of the Galois symbol $K_2^M(K)/m\\rightarrow H^2(K,\\mu_m^{\\otimes 2})$ of the Bloch-Kato conjecture, where $K_2^M(K)$ is the Milnor $K$-group of $K$. In this paper we give a geometric description of the image of this generalized Galois symbol by looking at the Tate duality pairing $H^{2}(K,A["},"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":"1703.06974","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-03-20T21:25:22Z","cross_cats_sorted":[],"title_canon_sha256":"c70a3d08f70e2d5d8bd32994626d17d2bdebeb0c728451afd40991fc1ee376ac","abstract_canon_sha256":"ea198ad8de1eee57c72d53135597b06ec1b9a2670aa338297ded5b6ec6f91cd3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:16:50.816399Z","signature_b64":"eX8PIYeMITifwhJNh8KsR0Ru4vPBnD0oeuvST5bgenj1IW/mv3exTJg+Up8wVTChxRJbWw7bT3amzsRRnriaCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"91cf2669a033f74207a3bbcc57899bc223609b9211d9115722bef4f8b8feaa19","last_reissued_at":"2026-05-18T00:16:50.815768Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:16:50.815768Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"A finer Tate duality theorem for local Galois symbols","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Evangelia Gazaki","submitted_at":"2017-03-20T21:25:22Z","abstract_excerpt":"Let $K$ be a finite extension of $\\mathbb{Q}_p$. 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