{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2018:K3SO4523M5DFWKEWQ23BDVUJY7","short_pith_number":"pith:K3SO4523","canonical_record":{"source":{"id":"1805.00131","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-04-30T23:31:05Z","cross_cats_sorted":[],"title_canon_sha256":"82acca03b1a2ef1a51a1f8f1c3442dda103898a7368dff874ab863674c8c8cc1","abstract_canon_sha256":"25a61b4adcd2a51bb7e8cb8e3d91d46ed42a4f4c5d13009ebaa41a47d4a2e638"},"schema_version":"1.0"},"canonical_sha256":"56e4ee775b67465b289686b611d689c7eba8976c23088d4a34a32a805b191008","source":{"kind":"arxiv","id":"1805.00131","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.00131","created_at":"2026-05-18T00:13:44Z"},{"alias_kind":"arxiv_version","alias_value":"1805.00131v2","created_at":"2026-05-18T00:13:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.00131","created_at":"2026-05-18T00:13:44Z"},{"alias_kind":"pith_short_12","alias_value":"K3SO4523M5DF","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_16","alias_value":"K3SO4523M5DFWKEW","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_8","alias_value":"K3SO4523","created_at":"2026-05-18T12:32:33Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2018:K3SO4523M5DFWKEWQ23BDVUJY7","target":"record","payload":{"canonical_record":{"source":{"id":"1805.00131","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-04-30T23:31:05Z","cross_cats_sorted":[],"title_canon_sha256":"82acca03b1a2ef1a51a1f8f1c3442dda103898a7368dff874ab863674c8c8cc1","abstract_canon_sha256":"25a61b4adcd2a51bb7e8cb8e3d91d46ed42a4f4c5d13009ebaa41a47d4a2e638"},"schema_version":"1.0"},"canonical_sha256":"56e4ee775b67465b289686b611d689c7eba8976c23088d4a34a32a805b191008","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:13:44.330507Z","signature_b64":"E4UsL6uVms7DY+dJCOBdbwx5yIwjo49oZ27yhk1n7orVCcCFLNNMeA975pQPh/vMGwizgMTYfeipd/Z8IcTcAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"56e4ee775b67465b289686b611d689c7eba8976c23088d4a34a32a805b191008","last_reissued_at":"2026-05-18T00:13:44.329830Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:13:44.329830Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1805.00131","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T00:13:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"iil+SXFGzMt9DN4Mibqkjhf+JDIdmAP1mWZIHcj8CLzT176XiniyGpiiHsKklcyvig9LfFN7l9u466mUQq+xAA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T20:56:41.236405Z"},"content_sha256":"b8ea64fd80652e0e30188a1fc85bd492341b638220054d165ec0ffe63f7e6582","schema_version":"1.0","event_id":"sha256:b8ea64fd80652e0e30188a1fc85bd492341b638220054d165ec0ffe63f7e6582"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2018:K3SO4523M5DFWKEWQ23BDVUJY7","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Wieferich Primes and a mod $p$ Leopoldt Conjecture","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chandrashekhar Khare, David-A. Guiraud, Gebhard Boeckle, Sudesh Kalyanswamy","submitted_at":"2018-04-30T23:31:05Z","abstract_excerpt":"We consider questions in Galois cohomology which arise by considering mod $p$ Galois representations arising from automorphic forms. We consider a Galois cohomological analog for the standard heuristics about the distribution of Wieferich primes, i.e. prime $p$ such that $2^{p-1}$ is 1 mod $p^2$. Our analog relates to asking if in a compatible system of Galois representations, for almost all primes $p$, the residual mod $p$ representation arising from it has unobstructed deformation theory. This analog leads in particular to formulating a mod $p$ analog for almost all primes $p$ of the classic"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.00131","kind":"arxiv","version":2},"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":"2026-05-18T00:13:44Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MOu4g0cy2FaO1uZ+hCWPjzgKyyoijl/bGuRstQzshwB0SqcLxANMidy9PhYtNBhUEsLR36/CH1V2lZWp2quHCg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-30T20:56:41.236763Z"},"content_sha256":"83a1d5c1bc40429f750d411ea11743719b476320576c9c27d4a12bc4f5945b66","schema_version":"1.0","event_id":"sha256:83a1d5c1bc40429f750d411ea11743719b476320576c9c27d4a12bc4f5945b66"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/K3SO4523M5DFWKEWQ23BDVUJY7/bundle.json","state_url":"https://pith.science/pith/K3SO4523M5DFWKEWQ23BDVUJY7/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/K3SO4523M5DFWKEWQ23BDVUJY7/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":"2026-06-30T20:56:41Z","links":{"resolver":"https://pith.science/pith/K3SO4523M5DFWKEWQ23BDVUJY7","bundle":"https://pith.science/pith/K3SO4523M5DFWKEWQ23BDVUJY7/bundle.json","state":"https://pith.science/pith/K3SO4523M5DFWKEWQ23BDVUJY7/state.json","well_known_bundle":"https://pith.science/.well-known/pith/K3SO4523M5DFWKEWQ23BDVUJY7/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2018:K3SO4523M5DFWKEWQ23BDVUJY7","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":"25a61b4adcd2a51bb7e8cb8e3d91d46ed42a4f4c5d13009ebaa41a47d4a2e638","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-04-30T23:31:05Z","title_canon_sha256":"82acca03b1a2ef1a51a1f8f1c3442dda103898a7368dff874ab863674c8c8cc1"},"schema_version":"1.0","source":{"id":"1805.00131","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1805.00131","created_at":"2026-05-18T00:13:44Z"},{"alias_kind":"arxiv_version","alias_value":"1805.00131v2","created_at":"2026-05-18T00:13:44Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1805.00131","created_at":"2026-05-18T00:13:44Z"},{"alias_kind":"pith_short_12","alias_value":"K3SO4523M5DF","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_16","alias_value":"K3SO4523M5DFWKEW","created_at":"2026-05-18T12:32:33Z"},{"alias_kind":"pith_short_8","alias_value":"K3SO4523","created_at":"2026-05-18T12:32:33Z"}],"graph_snapshots":[{"event_id":"sha256:83a1d5c1bc40429f750d411ea11743719b476320576c9c27d4a12bc4f5945b66","target":"graph","created_at":"2026-05-18T00:13:44Z","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":"We consider questions in Galois cohomology which arise by considering mod $p$ Galois representations arising from automorphic forms. We consider a Galois cohomological analog for the standard heuristics about the distribution of Wieferich primes, i.e. prime $p$ such that $2^{p-1}$ is 1 mod $p^2$. Our analog relates to asking if in a compatible system of Galois representations, for almost all primes $p$, the residual mod $p$ representation arising from it has unobstructed deformation theory. This analog leads in particular to formulating a mod $p$ analog for almost all primes $p$ of the classic","authors_text":"Chandrashekhar Khare, David-A. Guiraud, Gebhard Boeckle, Sudesh Kalyanswamy","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-04-30T23:31:05Z","title":"Wieferich Primes and a mod $p$ Leopoldt Conjecture"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1805.00131","kind":"arxiv","version":2},"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:b8ea64fd80652e0e30188a1fc85bd492341b638220054d165ec0ffe63f7e6582","target":"record","created_at":"2026-05-18T00:13:44Z","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":"25a61b4adcd2a51bb7e8cb8e3d91d46ed42a4f4c5d13009ebaa41a47d4a2e638","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2018-04-30T23:31:05Z","title_canon_sha256":"82acca03b1a2ef1a51a1f8f1c3442dda103898a7368dff874ab863674c8c8cc1"},"schema_version":"1.0","source":{"id":"1805.00131","kind":"arxiv","version":2}},"canonical_sha256":"56e4ee775b67465b289686b611d689c7eba8976c23088d4a34a32a805b191008","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"56e4ee775b67465b289686b611d689c7eba8976c23088d4a34a32a805b191008","first_computed_at":"2026-05-18T00:13:44.329830Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:13:44.329830Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"E4UsL6uVms7DY+dJCOBdbwx5yIwjo49oZ27yhk1n7orVCcCFLNNMeA975pQPh/vMGwizgMTYfeipd/Z8IcTcAg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:13:44.330507Z","signed_message":"canonical_sha256_bytes"},"source_id":"1805.00131","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:b8ea64fd80652e0e30188a1fc85bd492341b638220054d165ec0ffe63f7e6582","sha256:83a1d5c1bc40429f750d411ea11743719b476320576c9c27d4a12bc4f5945b66"],"state_sha256":"3e06aedffaf0cbc0ee38aea8322d77339ccc6f52cfb9fee81174f5978aa960ef"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"J2eKlyA0/fb+fd8LdBq5jmlAdC5+mpalgb5sbzc2kci7ODTIcc9W6kvlKzHspxaqedK783dBdJHOe24v5l8DAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-30T20:56:41.238719Z","bundle_sha256":"a530e94fe48996ce726ed0f5f25ed94760a931bbed1d56a042277a77a0f80609"}}