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When E/Q has analytic rank 2 and E/K has analytic rank 3, shadow lines are expected to lie in E(Q)\\otimes Q_p. If, in addition, p splits in K/Q, then shadow lines can be determined using the anticyclotomic p-adic height pairing. We develop an algorithm to compute anticyclotomic p-adic heights which we then use to provide an algorithm to compute shadow lines. We con"},"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":"1610.08729","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-10-27T12:03:13Z","cross_cats_sorted":[],"title_canon_sha256":"f42ebd859065350722de8c528888ba67dd03ebf98399c15bfcaaf20d8346e613","abstract_canon_sha256":"b83eeaba8369081015a88cb1f4a1661066d606d1163c4ad8e3718fdbf8b2e4ed"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:01:07.418426Z","signature_b64":"Uxf1jvBg4PLah2mk1dBCmX8xBjwEyE0r2y8x9awF2up5n+pUx1xfo5sGOwWfbEC92IJYC+zLUV43pS83mFH2BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"999d5154e44502f4066525efbe4612bd89ef2a326e6cc3a2c43bd4407116b6f3","last_reissued_at":"2026-05-18T01:01:07.417773Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:01:07.417773Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Shadow lines in the arithmetic of elliptic curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Bahare Mirza, Jaclyn Lang, Jennifer S. Balakrishnan, Mirela Ciperiani, Rachel Newton","submitted_at":"2016-10-27T12:03:13Z","abstract_excerpt":"Let E/Q be an elliptic curve and p a rational prime of good ordinary reduction. For every imaginary quadratic field K/Q satisfying the Heegner hypothesis for E we have a corresponding line in E(K)\\otimes Q_p, known as a shadow line. When E/Q has analytic rank 2 and E/K has analytic rank 3, shadow lines are expected to lie in E(Q)\\otimes Q_p. If, in addition, p splits in K/Q, then shadow lines can be determined using the anticyclotomic p-adic height pairing. We develop an algorithm to compute anticyclotomic p-adic heights which we then use to provide an algorithm to compute shadow lines. 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