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We classify the possible images for the global Galois representation in the case where $G$ is a Cartan subgroup or the normalizer of a Cartan subgroup. When $K = \\mathbf{Q}$, we deduce a counterexample to the local-global principle in the case where $G$ is the normalizer of a split Cartan and $\\ell = 13$. In particular, there are at least three elliptic curves (up to twist) over $\\mathbf{Q}$ whose mod $13$ image of Galois is locally co"},"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":"1502.01288","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-02-04T18:49:37Z","cross_cats_sorted":[],"title_canon_sha256":"25ed683eb4a4d1c40a8d38fb034be351e4f83b27a98d3b1ec28fd0422ab51aa0","abstract_canon_sha256":"cbc4eac11b24bf06685a2b7255f0520a61f46d97f16ff34fe83d26f5c21baaef"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:27:57.636887Z","signature_b64":"cQ6G/IrekJ8kbb5UKsy0Mw3fv6xWcbN9x80DfQM9DIJxoo5EBr3JZEY2edNZB0o5wTUHNu4O9qN/6bkurWgwDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"0a0a215762aa5a34e940b1f00dcbda955cbb2ea62d38d729709ac869d0bbc3eb","last_reissued_at":"2026-05-18T02:27:57.636440Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:27:57.636440Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Local-Global principles for certain images of Galois representations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Anastassia Etropolski","submitted_at":"2015-02-04T18:49:37Z","abstract_excerpt":"Let $K$ be a number field and let $E/K$ be an elliptic curve whose mod $\\ell$ Galois representation locally has image contained in a group $G$, up to conjugacy. We classify the possible images for the global Galois representation in the case where $G$ is a Cartan subgroup or the normalizer of a Cartan subgroup. When $K = \\mathbf{Q}$, we deduce a counterexample to the local-global principle in the case where $G$ is the normalizer of a split Cartan and $\\ell = 13$. 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