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The first result is that for all but a finite number of primes p where E has ordinary reduction, the image of T_F(A)/p in the Galois cohomology group H^2(F, sym^2(E[p])) is zero; here E[p] denotes as usual the Galois module of p-division points on E. The second result is that for any prime p where E has good ordinary reduction, there is a finite extension K of F, "},"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":"2407.20468","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2024-07-29T23:59:29Z","cross_cats_sorted":["math.AG"],"title_canon_sha256":"4f95ad1a3b68c5a3e10056f2bd7ca877bdbf69d1a065145867e60f8e17adae5e","abstract_canon_sha256":"70f83ef29ec81bff4c08effa4aa6fb54eec576957ee05263442fee6ebed003a2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T09:37:55.767440Z","signature_b64":"4Fox5CDpYNd0TIV/e44a3tgDY458YjYkPAcR8O6t4SZu3yru9MMuNK2DH98yKPBXPSMQb63BucQW+bolFTGXCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"853f6ae0129cbcb48efc65b4129af68d901f430b9c58da58dfb88dabaaa66ec5","last_reissued_at":"2026-07-05T09:37:55.766950Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T09:37:55.766950Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Global Galois Symbols on E x E","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG"],"primary_cat":"math.NT","authors_text":"Dinakar Ramakrishnan","submitted_at":"2024-07-29T23:59:29Z","abstract_excerpt":"Let E be an elliptic curve over a number field F, A the abelian surface E x E, and T_F(A) the F-rational albanese kernel of A, which is a subgroup of the degree zero part of Chow group of zero cycles on A modulo rational equivalence. The first result is that for all but a finite number of primes p where E has ordinary reduction, the image of T_F(A)/p in the Galois cohomology group H^2(F, sym^2(E[p])) is zero; here E[p] denotes as usual the Galois module of p-division points on E. 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