{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:V2XRKLIYLT6DGCDH2MZUGTNDFK","short_pith_number":"pith:V2XRKLIY","canonical_record":{"source":{"id":"1606.06535","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-06-21T12:28:47Z","cross_cats_sorted":[],"title_canon_sha256":"c4ab6125983ca166254378f5abcb68e8be960b86322b8acd3d4e6ad69e258fd7","abstract_canon_sha256":"7ba68e468d4c283c3c64621622848762fd6133b608bb1546ea5bd949c9c40d6b"},"schema_version":"1.0"},"canonical_sha256":"aeaf152d185cfc330867d333434da32aa59202b03c38253045af8dad4923fc4d","source":{"kind":"arxiv","id":"1606.06535","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.06535","created_at":"2026-05-18T01:12:09Z"},{"alias_kind":"arxiv_version","alias_value":"1606.06535v1","created_at":"2026-05-18T01:12:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.06535","created_at":"2026-05-18T01:12:09Z"},{"alias_kind":"pith_short_12","alias_value":"V2XRKLIYLT6D","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"V2XRKLIYLT6DGCDH","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"V2XRKLIY","created_at":"2026-05-18T12:30:46Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:V2XRKLIYLT6DGCDH2MZUGTNDFK","target":"record","payload":{"canonical_record":{"source":{"id":"1606.06535","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-06-21T12:28:47Z","cross_cats_sorted":[],"title_canon_sha256":"c4ab6125983ca166254378f5abcb68e8be960b86322b8acd3d4e6ad69e258fd7","abstract_canon_sha256":"7ba68e468d4c283c3c64621622848762fd6133b608bb1546ea5bd949c9c40d6b"},"schema_version":"1.0"},"canonical_sha256":"aeaf152d185cfc330867d333434da32aa59202b03c38253045af8dad4923fc4d","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:12:09.305360Z","signature_b64":"ekTQCm2zWlgWaePRwslKKct5sdDcqCqiEaetPhS32AXsd7Z99tVzTYtiFhCjeXxDb00Zy9c6C8SZYm7CCUkNCA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"aeaf152d185cfc330867d333434da32aa59202b03c38253045af8dad4923fc4d","last_reissued_at":"2026-05-18T01:12:09.304933Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:12:09.304933Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1606.06535","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:12:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"jTWjzWE5n3dhi16spGzfOv6IJcg/FWJHvTTB4sghCo4rAYz4CnNFeTlTd3w/0Dx0iPgTp+8t5K1oOrD6gc4UBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T12:42:03.957607Z"},"content_sha256":"f2094b4c99128e9a8034c441fac0302b671a7093d1f0dc2becb1307760822aac","schema_version":"1.0","event_id":"sha256:f2094b4c99128e9a8034c441fac0302b671a7093d1f0dc2becb1307760822aac"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:V2XRKLIYLT6DGCDH2MZUGTNDFK","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On lifting and modularity of reducible residual Galois representations over imaginary quadratic fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Krzysztof Klosin, Tobias Berger","submitted_at":"2016-06-21T12:28:47Z","abstract_excerpt":"In this paper we study deformations of mod $p$ Galois representations $\\tau$ (over an imaginary quadratic field $F$) of dimension $2$ whose semi-simplification is the direct sum of two characters $\\tau_1$ and $\\tau_2$. As opposed to our previous work we do not impose any restrictions on the dimension of the crystalline Selmer group $H^1_{\\Sigma}(F, {\\rm Hom}(\\tau_2, \\tau_1)) \\subset {\\rm Ext}^1(\\tau_2, \\tau_1)$. We establish that there exists a basis $\\mathcal{B}$ of $H^1_{\\Sigma}(F, {\\rm Hom}(\\tau_2, \\tau_1))$ arising from automorphic representations over $F$ (Theorem 8.1). Assuming among oth"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.06535","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"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T01:12:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"rjn7RwdKwVeSe/lo2Hd3FC334MZDRpGZwgNDT+/uYIHaIGTxauLqs8yIKSxjvHw+PW08exU1qYIRC3ErXgW1Cg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-07T12:42:03.958237Z"},"content_sha256":"1d35d8b517164b52d2f44de745bd4926497c0561b9adcaeea12cce79276e50c4","schema_version":"1.0","event_id":"sha256:1d35d8b517164b52d2f44de745bd4926497c0561b9adcaeea12cce79276e50c4"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/V2XRKLIYLT6DGCDH2MZUGTNDFK/bundle.json","state_url":"https://pith.science/pith/V2XRKLIYLT6DGCDH2MZUGTNDFK/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/V2XRKLIYLT6DGCDH2MZUGTNDFK/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-06-07T12:42:03Z","links":{"resolver":"https://pith.science/pith/V2XRKLIYLT6DGCDH2MZUGTNDFK","bundle":"https://pith.science/pith/V2XRKLIYLT6DGCDH2MZUGTNDFK/bundle.json","state":"https://pith.science/pith/V2XRKLIYLT6DGCDH2MZUGTNDFK/state.json","well_known_bundle":"https://pith.science/.well-known/pith/V2XRKLIYLT6DGCDH2MZUGTNDFK/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:V2XRKLIYLT6DGCDH2MZUGTNDFK","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"7ba68e468d4c283c3c64621622848762fd6133b608bb1546ea5bd949c9c40d6b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-06-21T12:28:47Z","title_canon_sha256":"c4ab6125983ca166254378f5abcb68e8be960b86322b8acd3d4e6ad69e258fd7"},"schema_version":"1.0","source":{"id":"1606.06535","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1606.06535","created_at":"2026-05-18T01:12:09Z"},{"alias_kind":"arxiv_version","alias_value":"1606.06535v1","created_at":"2026-05-18T01:12:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1606.06535","created_at":"2026-05-18T01:12:09Z"},{"alias_kind":"pith_short_12","alias_value":"V2XRKLIYLT6D","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_16","alias_value":"V2XRKLIYLT6DGCDH","created_at":"2026-05-18T12:30:46Z"},{"alias_kind":"pith_short_8","alias_value":"V2XRKLIY","created_at":"2026-05-18T12:30:46Z"}],"graph_snapshots":[{"event_id":"sha256:1d35d8b517164b52d2f44de745bd4926497c0561b9adcaeea12cce79276e50c4","target":"graph","created_at":"2026-05-18T01:12:09Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"paper":{"abstract_excerpt":"In this paper we study deformations of mod $p$ Galois representations $\\tau$ (over an imaginary quadratic field $F$) of dimension $2$ whose semi-simplification is the direct sum of two characters $\\tau_1$ and $\\tau_2$. As opposed to our previous work we do not impose any restrictions on the dimension of the crystalline Selmer group $H^1_{\\Sigma}(F, {\\rm Hom}(\\tau_2, \\tau_1)) \\subset {\\rm Ext}^1(\\tau_2, \\tau_1)$. We establish that there exists a basis $\\mathcal{B}$ of $H^1_{\\Sigma}(F, {\\rm Hom}(\\tau_2, \\tau_1))$ arising from automorphic representations over $F$ (Theorem 8.1). Assuming among oth","authors_text":"Krzysztof Klosin, Tobias Berger","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-06-21T12:28:47Z","title":"On lifting and modularity of reducible residual Galois representations over imaginary quadratic fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1606.06535","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:f2094b4c99128e9a8034c441fac0302b671a7093d1f0dc2becb1307760822aac","target":"record","created_at":"2026-05-18T01:12:09Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"7ba68e468d4c283c3c64621622848762fd6133b608bb1546ea5bd949c9c40d6b","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-06-21T12:28:47Z","title_canon_sha256":"c4ab6125983ca166254378f5abcb68e8be960b86322b8acd3d4e6ad69e258fd7"},"schema_version":"1.0","source":{"id":"1606.06535","kind":"arxiv","version":1}},"canonical_sha256":"aeaf152d185cfc330867d333434da32aa59202b03c38253045af8dad4923fc4d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"aeaf152d185cfc330867d333434da32aa59202b03c38253045af8dad4923fc4d","first_computed_at":"2026-05-18T01:12:09.304933Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:12:09.304933Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ekTQCm2zWlgWaePRwslKKct5sdDcqCqiEaetPhS32AXsd7Z99tVzTYtiFhCjeXxDb00Zy9c6C8SZYm7CCUkNCA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:12:09.305360Z","signed_message":"canonical_sha256_bytes"},"source_id":"1606.06535","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:f2094b4c99128e9a8034c441fac0302b671a7093d1f0dc2becb1307760822aac","sha256:1d35d8b517164b52d2f44de745bd4926497c0561b9adcaeea12cce79276e50c4"],"state_sha256":"8ef9fd133f1c65bca6defd2ce1bde24ae0d98ff0a869acb9ccbb44548f7118e3"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xCXnoEwMRwx4/itq9LJqWsB2zn8DXVTYsNQ3MuTYf1Vx4OjdhcvEVo8i/mQ/w2ME5uflUoz9kMR+bNxt+Zd/Aw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-07T12:42:03.961616Z","bundle_sha256":"874dbe06d574026915ce4fa489495432d9cb9124c702f5f113903665f7761372"}}