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Namely, we show if the difference of the $2$-Selmer ranks of $E^\\chi$ and $A^\\chi$ is bounded independent of $\\chi$, there is a $G_K$-isomorphism $E[2] \\cong A[2]$."},"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":"1610.01195","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-10-04T20:40:01Z","cross_cats_sorted":[],"title_canon_sha256":"285944cbff01ad988bb392f3d2fa980a7efc55be5f7e4f9b797493f3964d52d1","abstract_canon_sha256":"c940ffcbc169378fea566fdfc7b9b55570a61c8038d50be4800948813d1abd04"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:59:57.046946Z","signature_b64":"DRzraFZ1RhQCUCI11DTCyaY9HkzB2dVaAepER6lJV7v7R2h+eGk6PV2T4ASiXdl73zRFy/7Nz1DnY1QCnJlpDA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"27a9aeebdf80e6e614539dbd4e58dc1570c968d6cc590230d19826caed3d1f43","last_reissued_at":"2026-05-18T00:59:57.046522Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:59:57.046522Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"2-Selmer near-companion curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Myungjun Yu","submitted_at":"2016-10-04T20:40:01Z","abstract_excerpt":"Let $E$ and $A$ be elliptic curves over a number field $K$. 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