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Given two elliptic curves $E_1$ and $E_2$ over a number field $K$, Mazur-Rubin\\cite{mr} have defined them to be {\\it $n$-Selmer companion} if for every quadratic twist $\\chi$ of $K$, the $n$-Selmer groups of $E_1^\\chi $ and $E_2^\\chi$ over $K$ are isomorphic. Given a prime $p$, they have given sufficient conditions for two elliptic curves to be $p^r$-Selmer companion in terms of mod-$p^r$ congruences between the curves. We discuss "},"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":"1806.04944","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-06-13T10:51:10Z","cross_cats_sorted":[],"title_canon_sha256":"46e346862de1b24a7f9ba744c89dd3515c0241913adeecbab323f80be994e3fc","abstract_canon_sha256":"71868bfcc2b2473a1c6169b890cef120d6c0ad2abad66ddc0515a30d2a7445f2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-17T23:56:29.773124Z","signature_b64":"hBK8M3fnw1UM+qlXsoVegFzlHPe0s+8z8P0eIxPUMHZ1DUlg8QfRqMFcqr7LapCqVoUNTThEHreB3ELwJD5PCg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"5c93a44c097c93731bb0b4a977f75d55bafb157e28041147a4e37c5f0e2b9d38","last_reissued_at":"2026-05-17T23:56:29.772719Z","signature_status":"signed_v1","first_computed_at":"2026-05-17T23:56:29.772719Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"$p^r$-Selmer companion modular forms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Dipramit Majumdar, Somnath Jha, Sudhanshu Shekhar","submitted_at":"2018-06-13T10:51:10Z","abstract_excerpt":"The study of $n$-Selmer group of elliptic curve over number field in recent past has led to the discovery of some deep results in the arithmetic of elliptic curves. 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