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The size of this module is roughly measured by its rank $\\tau$ over a $p$-adic Galois group algebra $\\Lambda(H)$, which has been studied in the past decade. We prove $\\tau >= 2$ for almost every elliptic curve under standard assumptions. Following from a result of Coates et al, $\\tau$ is odd if and only if $[Q(E[p]) \\colon Q"},"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":"1303.0710","kind":"arxiv","version":4},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-03-04T14:35:10Z","cross_cats_sorted":[],"title_canon_sha256":"19693449d872dab207f0c2775eb1c75c57e339e5c56e44cf7482f8cd0924635c","abstract_canon_sha256":"39c61ea7e3dde81896119d6a79c6c28d7d02521f95a0afe5fc404b77839752e9"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:17:56.692694Z","signature_b64":"gWFwuM1eleXc7n3T8X3Eh3r6Wn4Ztx1o5tgQtz/qB9VljHAz1rINAwGrHu6q9icTd6vlwRZaxTfotS9KTmfBCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8c1c675f3782d25a2ad3712be72309ce492ff6336d80c43957fb0e4489ca064d","last_reissued_at":"2026-05-18T03:17:56.692010Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:17:56.692010Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Ranks of GL2 Iwasawa modules of elliptic curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Tibor Backhausz","submitted_at":"2013-03-04T14:35:10Z","abstract_excerpt":"Let $p >= 5$ be a prime and $E$ an elliptic curve without complex multiplication and let $K_\\infty=Q(E[p^\\infty])$ be a pro-$p$ Galois extension over a number field $K$. 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