{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2018:7DZOSHLMHIMN4XDMZXVGTYH4SW","short_pith_number":"pith:7DZOSHLM","schema_version":"1.0","canonical_sha256":"f8f2e91d6c3a18de5c6ccdea69e0fc95ba2173cfb7a6dfb4a12751d7170432e2","source":{"kind":"arxiv","id":"1810.07808","version":1},"attestation_state":"computed","paper":{"title":"Modularity of residual Galois extensions and the Eisenstein ideal","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Krzysztof Klosin, Tobias Berger","submitted_at":"2018-10-17T21:16:51Z","abstract_excerpt":"For a totally real field $F$, a finite extension $\\mathbf{F}$ of $\\mathbf{F}_p$ and a Galois character $\\chi: G_F \\to \\mathbf{F}^{\\times}$ unramified away from a finite set of places $\\Sigma \\supset \\{\\mathfrak{p} \\mid p\\}$ consider the Bloch-Kato Selmer group $H:=H^1_{\\Sigma}(F, \\chi^{-1})$. In an earlier paper of the authors it was proved that the number $d$ of isomorphism classes of (non-semisimple, reducible) residual representations $\\overline{\\rho}$ giving rise to lines in $H$ which are modular by some $\\rho_f$ (also unramified outside $\\Sigma$) satisfies $d \\geq n:= \\dim_{\\mathbf{F}} H$"},"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":"1810.07808","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-10-17T21:16:51Z","cross_cats_sorted":[],"title_canon_sha256":"63fcd16d27ee27dda48403b4a9b7f3caa7cd20734b3d563ec7b90763e104143a","abstract_canon_sha256":"0d4e93fd743670a835a6624d8f331e91d1800ad2d7955b544c775b81d88675ff"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:02:54.149697Z","signature_b64":"6xzJ20Eu8J/xfZymVe7M2XLYKrTL+7LxaPwSEYyfnaChkwnRsla1p2p3NcPXPez2C+E5JjWg9MRNCK5Mu0OoDw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f8f2e91d6c3a18de5c6ccdea69e0fc95ba2173cfb7a6dfb4a12751d7170432e2","last_reissued_at":"2026-05-18T00:02:54.149063Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:02:54.149063Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Modularity of residual Galois extensions and the Eisenstein ideal","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Krzysztof Klosin, Tobias Berger","submitted_at":"2018-10-17T21:16:51Z","abstract_excerpt":"For a totally real field $F$, a finite extension $\\mathbf{F}$ of $\\mathbf{F}_p$ and a Galois character $\\chi: G_F \\to \\mathbf{F}^{\\times}$ unramified away from a finite set of places $\\Sigma \\supset \\{\\mathfrak{p} \\mid p\\}$ consider the Bloch-Kato Selmer group $H:=H^1_{\\Sigma}(F, \\chi^{-1})$. In an earlier paper of the authors it was proved that the number $d$ of isomorphism classes of (non-semisimple, reducible) residual representations $\\overline{\\rho}$ giving rise to lines in $H$ which are modular by some $\\rho_f$ (also unramified outside $\\Sigma$) satisfies $d \\geq n:= \\dim_{\\mathbf{F}} H$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1810.07808","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"aliases":[{"alias_kind":"arxiv","alias_value":"1810.07808","created_at":"2026-05-18T00:02:54.149150+00:00"},{"alias_kind":"arxiv_version","alias_value":"1810.07808v1","created_at":"2026-05-18T00:02:54.149150+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1810.07808","created_at":"2026-05-18T00:02:54.149150+00:00"},{"alias_kind":"pith_short_12","alias_value":"7DZOSHLMHIMN","created_at":"2026-05-18T12:32:11.075285+00:00"},{"alias_kind":"pith_short_16","alias_value":"7DZOSHLMHIMN4XDM","created_at":"2026-05-18T12:32:11.075285+00:00"},{"alias_kind":"pith_short_8","alias_value":"7DZOSHLM","created_at":"2026-05-18T12:32:11.075285+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/7DZOSHLMHIMN4XDMZXVGTYH4SW","json":"https://pith.science/pith/7DZOSHLMHIMN4XDMZXVGTYH4SW.json","graph_json":"https://pith.science/api/pith-number/7DZOSHLMHIMN4XDMZXVGTYH4SW/graph.json","events_json":"https://pith.science/api/pith-number/7DZOSHLMHIMN4XDMZXVGTYH4SW/events.json","paper":"https://pith.science/paper/7DZOSHLM"},"agent_actions":{"view_html":"https://pith.science/pith/7DZOSHLMHIMN4XDMZXVGTYH4SW","download_json":"https://pith.science/pith/7DZOSHLMHIMN4XDMZXVGTYH4SW.json","view_paper":"https://pith.science/paper/7DZOSHLM","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1810.07808&json=true","fetch_graph":"https://pith.science/api/pith-number/7DZOSHLMHIMN4XDMZXVGTYH4SW/graph.json","fetch_events":"https://pith.science/api/pith-number/7DZOSHLMHIMN4XDMZXVGTYH4SW/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/7DZOSHLMHIMN4XDMZXVGTYH4SW/action/timestamp_anchor","attest_storage":"https://pith.science/pith/7DZOSHLMHIMN4XDMZXVGTYH4SW/action/storage_attestation","attest_author":"https://pith.science/pith/7DZOSHLMHIMN4XDMZXVGTYH4SW/action/author_attestation","sign_citation":"https://pith.science/pith/7DZOSHLMHIMN4XDMZXVGTYH4SW/action/citation_signature","submit_replication":"https://pith.science/pith/7DZOSHLMHIMN4XDMZXVGTYH4SW/action/replication_record"}},"created_at":"2026-05-18T00:02:54.149150+00:00","updated_at":"2026-05-18T00:02:54.149150+00:00"}