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We study Sel_A(K)_l^\\vee (the Pontrjagin dual of the Selmer group) and (under some mild hypotheses) prove that it is a finitely generated Z_l[[\\Gal(K/F)]]-module via generalizations of Mazur's Control Theorem. If Gal(K/F) has no elements of order l and contains a closed normal subgroup H such that Gal(K/F)/H\\simeq Z_l, we are able to give sufficient conditions for Sel_A(K)_l^\\vee to be finitely generated as Z_"},"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":"1206.2767","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2012-06-13T11:02:48Z","cross_cats_sorted":[],"title_canon_sha256":"f4884d914d3ba1ba2b3280e40cef4c5d581c540500a864a6fdda0e6bc8734ffe","abstract_canon_sha256":"a026c2379c2a5c99cd8d8be67176c1da968f2e764ed6d48ab5b48534ce5499a3"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:29:03.923342Z","signature_b64":"lSZxd8CNBiucn27S87oipjn7NcaqTKGNoXpJduCIaX3ITb82+X6iB/IWPxv+AdtgwUOUG0zaoycYkQXrJTp1Aw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"276e0328ccec5bc3c9755ff47c6fbfc954d9f241fcd9c7937c3d0cd79735a7b7","last_reissued_at":"2026-05-18T03:29:03.922920Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:29:03.922920Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Control Theorems for l-adic Lie extensions of global function fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andrea Bandini, Maria Valentino","submitted_at":"2012-06-13T11:02:48Z","abstract_excerpt":"Let F be a global function field of characteristic p>0, K/F an l-adic Lie extension unramified outside a finite set of places S and A/F an abelian variety without complex multiplication. 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