{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:5Z54YYBT2NC6RBZX5WGMAOGWRG","short_pith_number":"pith:5Z54YYBT","canonical_record":{"source":{"id":"1109.2440","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-09-12T12:21:56Z","cross_cats_sorted":[],"title_canon_sha256":"5a68c27e3dae43196d79646e26f4406da82cb87f6d1bcb8ceaa70400033e7c60","abstract_canon_sha256":"338c0dcf28b4eab4964db456c869ce8d755c12036a4cbbda7471d75462373412"},"schema_version":"1.0"},"canonical_sha256":"ee7bcc6033d345e88737ed8cc038d6898df47b348a026b4e109f67b308d354f9","source":{"kind":"arxiv","id":"1109.2440","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.2440","created_at":"2026-05-18T04:13:37Z"},{"alias_kind":"arxiv_version","alias_value":"1109.2440v1","created_at":"2026-05-18T04:13:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.2440","created_at":"2026-05-18T04:13:37Z"},{"alias_kind":"pith_short_12","alias_value":"5Z54YYBT2NC6","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"5Z54YYBT2NC6RBZX","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"5Z54YYBT","created_at":"2026-05-18T12:26:22Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:5Z54YYBT2NC6RBZX5WGMAOGWRG","target":"record","payload":{"canonical_record":{"source":{"id":"1109.2440","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-09-12T12:21:56Z","cross_cats_sorted":[],"title_canon_sha256":"5a68c27e3dae43196d79646e26f4406da82cb87f6d1bcb8ceaa70400033e7c60","abstract_canon_sha256":"338c0dcf28b4eab4964db456c869ce8d755c12036a4cbbda7471d75462373412"},"schema_version":"1.0"},"canonical_sha256":"ee7bcc6033d345e88737ed8cc038d6898df47b348a026b4e109f67b308d354f9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:13:37.659286Z","signature_b64":"zQWf3ezy0/YVi+TxkOLOlyLC+RjcfM+Qjim1HcsEKnVxpwLn32FVdcZ8e9qQfoHk6ERydjV8vLgmVQUcUUbJDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"ee7bcc6033d345e88737ed8cc038d6898df47b348a026b4e109f67b308d354f9","last_reissued_at":"2026-05-18T04:13:37.658781Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:13:37.658781Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1109.2440","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":"2026-05-18T04:13:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"+Fu5LSW4zRUEPwIRAATVqjc9eINrqhBjTb3GQDDkY4AIPZU9C/MXfyGx917L6w072rQC3CTCwINLXUpRE2+MDA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T15:36:50.671459Z"},"content_sha256":"2d93ed9630bf6dd957e4b4c773858e444935d4db1472d380cb86381dd3f2b906","schema_version":"1.0","event_id":"sha256:2d93ed9630bf6dd957e4b4c773858e444935d4db1472d380cb86381dd3f2b906"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:5Z54YYBT2NC6RBZX5WGMAOGWRG","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Radical characterizations of elliptic curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Antonella Perucca, Chris Hall","submitted_at":"2011-09-12T12:21:56Z","abstract_excerpt":"Let K be a number field, and let E be an elliptic curve over K. A famous result by Faltings of 1983 can be reformulated for elliptic curves as follows: if S is a set of primes of good reduction for E having density one, then the K-isogeny class of E is determined by the function which maps a prime in S to the size of the group of points over the residue field. In this paper, we prove that it suffices to look at the radical of the size."},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.2440","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":"2026-05-18T04:13:37Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ng/t++m/tlUJJopTyAQggjwLQR5tLRBufyh906dy82kcrwbqOD6/8NziYWvmTqk+6gM+1SrTwcT5TzAbKUj9AQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-23T15:36:50.671808Z"},"content_sha256":"1265226c0d336075987e1e0f800ea481712623f8fbc6f70fb32e9ccf9d0ab60d","schema_version":"1.0","event_id":"sha256:1265226c0d336075987e1e0f800ea481712623f8fbc6f70fb32e9ccf9d0ab60d"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/5Z54YYBT2NC6RBZX5WGMAOGWRG/bundle.json","state_url":"https://pith.science/pith/5Z54YYBT2NC6RBZX5WGMAOGWRG/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/5Z54YYBT2NC6RBZX5WGMAOGWRG/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":"2026-06-23T15:36:50Z","links":{"resolver":"https://pith.science/pith/5Z54YYBT2NC6RBZX5WGMAOGWRG","bundle":"https://pith.science/pith/5Z54YYBT2NC6RBZX5WGMAOGWRG/bundle.json","state":"https://pith.science/pith/5Z54YYBT2NC6RBZX5WGMAOGWRG/state.json","well_known_bundle":"https://pith.science/.well-known/pith/5Z54YYBT2NC6RBZX5WGMAOGWRG/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:5Z54YYBT2NC6RBZX5WGMAOGWRG","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":"338c0dcf28b4eab4964db456c869ce8d755c12036a4cbbda7471d75462373412","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-09-12T12:21:56Z","title_canon_sha256":"5a68c27e3dae43196d79646e26f4406da82cb87f6d1bcb8ceaa70400033e7c60"},"schema_version":"1.0","source":{"id":"1109.2440","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1109.2440","created_at":"2026-05-18T04:13:37Z"},{"alias_kind":"arxiv_version","alias_value":"1109.2440v1","created_at":"2026-05-18T04:13:37Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1109.2440","created_at":"2026-05-18T04:13:37Z"},{"alias_kind":"pith_short_12","alias_value":"5Z54YYBT2NC6","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_16","alias_value":"5Z54YYBT2NC6RBZX","created_at":"2026-05-18T12:26:22Z"},{"alias_kind":"pith_short_8","alias_value":"5Z54YYBT","created_at":"2026-05-18T12:26:22Z"}],"graph_snapshots":[{"event_id":"sha256:1265226c0d336075987e1e0f800ea481712623f8fbc6f70fb32e9ccf9d0ab60d","target":"graph","created_at":"2026-05-18T04:13:37Z","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":"Let K be a number field, and let E be an elliptic curve over K. A famous result by Faltings of 1983 can be reformulated for elliptic curves as follows: if S is a set of primes of good reduction for E having density one, then the K-isogeny class of E is determined by the function which maps a prime in S to the size of the group of points over the residue field. In this paper, we prove that it suffices to look at the radical of the size.","authors_text":"Antonella Perucca, Chris Hall","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-09-12T12:21:56Z","title":"Radical characterizations of elliptic curves"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1109.2440","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":"sha256:2d93ed9630bf6dd957e4b4c773858e444935d4db1472d380cb86381dd3f2b906","target":"record","created_at":"2026-05-18T04:13:37Z","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":"338c0dcf28b4eab4964db456c869ce8d755c12036a4cbbda7471d75462373412","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-09-12T12:21:56Z","title_canon_sha256":"5a68c27e3dae43196d79646e26f4406da82cb87f6d1bcb8ceaa70400033e7c60"},"schema_version":"1.0","source":{"id":"1109.2440","kind":"arxiv","version":1}},"canonical_sha256":"ee7bcc6033d345e88737ed8cc038d6898df47b348a026b4e109f67b308d354f9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"ee7bcc6033d345e88737ed8cc038d6898df47b348a026b4e109f67b308d354f9","first_computed_at":"2026-05-18T04:13:37.658781Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:13:37.658781Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"zQWf3ezy0/YVi+TxkOLOlyLC+RjcfM+Qjim1HcsEKnVxpwLn32FVdcZ8e9qQfoHk6ERydjV8vLgmVQUcUUbJDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:13:37.659286Z","signed_message":"canonical_sha256_bytes"},"source_id":"1109.2440","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:2d93ed9630bf6dd957e4b4c773858e444935d4db1472d380cb86381dd3f2b906","sha256:1265226c0d336075987e1e0f800ea481712623f8fbc6f70fb32e9ccf9d0ab60d"],"state_sha256":"53840b7a9d1f89bb7a3c4081fe82ad9adf1f5388e422397dc656640223a30e9d"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"6vGvVa00VUpKYw3cnrcuka2lX7gvRGIhP3ktn0wedUQS9y3LcNapxJcZdKTxq1SS4F2LzwnfOUw98wpaJ7B1Dw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-23T15:36:50.673886Z","bundle_sha256":"10fb6d2b449d7385eafc7b449e6f1a4aa23fbf74ff54aa7b75919c97e63cdbc9"}}