{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2017:HABCQY33QJ4HYN4JW3DOPY6EHZ","short_pith_number":"pith:HABCQY33","canonical_record":{"source":{"id":"1707.04804","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-07-16T01:32:17Z","cross_cats_sorted":[],"title_canon_sha256":"2d8094d4a69b833214ceaa3c615c727343a6c5569aa22b4c4456db8a1fe8428b","abstract_canon_sha256":"4125b3b89f07d84e7b85af5d6bfed66050e877417ce4e4fddc7dd6b279f5c881"},"schema_version":"1.0"},"canonical_sha256":"380228637b82787c3789b6c6e7e3c43e4573bfaaf50793a598b9ffa52ef0ab38","source":{"kind":"arxiv","id":"1707.04804","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.04804","created_at":"2026-05-18T00:40:12Z"},{"alias_kind":"arxiv_version","alias_value":"1707.04804v1","created_at":"2026-05-18T00:40:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.04804","created_at":"2026-05-18T00:40:12Z"},{"alias_kind":"pith_short_12","alias_value":"HABCQY33QJ4H","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_16","alias_value":"HABCQY33QJ4HYN4J","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_8","alias_value":"HABCQY33","created_at":"2026-05-18T12:31:18Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2017:HABCQY33QJ4HYN4JW3DOPY6EHZ","target":"record","payload":{"canonical_record":{"source":{"id":"1707.04804","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-07-16T01:32:17Z","cross_cats_sorted":[],"title_canon_sha256":"2d8094d4a69b833214ceaa3c615c727343a6c5569aa22b4c4456db8a1fe8428b","abstract_canon_sha256":"4125b3b89f07d84e7b85af5d6bfed66050e877417ce4e4fddc7dd6b279f5c881"},"schema_version":"1.0"},"canonical_sha256":"380228637b82787c3789b6c6e7e3c43e4573bfaaf50793a598b9ffa52ef0ab38","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:40:12.206291Z","signature_b64":"ELWZiWr1ygN2nA9OBMocwTAJGIH0hcL+y3UUYuGTMpEURedK/1yEwGRQfkf2jmCJ0ByXSR+zfJsJA0iv3QQEDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"380228637b82787c3789b6c6e7e3c43e4573bfaaf50793a598b9ffa52ef0ab38","last_reissued_at":"2026-05-18T00:40:12.205719Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:40:12.205719Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1707.04804","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":"2026-05-18T00:40:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"WS1LX73YByVsO/Zk2Cp40iUlyZ5vnkUt/qPnbwP5oNZRIfFgnX25PADm6clqHxxqgTJKccDcGAB7W8ik9FA/DA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T00:21:49.847729Z"},"content_sha256":"749a3f9cf93be3b0c02df2d84a1ee2aa6aa7b7e6cad0eaba3b1cf3a1ac07d78e","schema_version":"1.0","event_id":"sha256:749a3f9cf93be3b0c02df2d84a1ee2aa6aa7b7e6cad0eaba3b1cf3a1ac07d78e"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2017:HABCQY33QJ4HYN4JW3DOPY6EHZ","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Critical points of the classical Eisenstein series of weight two","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Chang-shou Lin, Zhijie Chen","submitted_at":"2017-07-16T01:32:17Z","abstract_excerpt":"In this paper, we completely determine the critical points of the normalized Eisenstein series $E_2(\\tau)$ of weight $2$. Although $E_2(\\tau)$ is not a modular form, our result shows that $E_2(\\tau)$ has at most one critical point in every fundamental domain of $\\Gamma_{0}(2)$. We also give a criteria for a fundamental domain containing a critical point of $E_2(\\tau)$. Furthermore, under the M\\\"obius transformation of $\\Gamma_{0}(2)$ action, all critical points can be mapped into the basic fundamental domain $F_0$ and their images are contained densely on three smooth curves. A geometric inter"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.04804","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":"2026-05-18T00:40:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"wHm5DZKpxE/3siJqa4P0U4O2p+vwX5bfpOU5/0ZmKQChkHAf8kEyj9XaMfBgmCwuCBUKw0nbabVxstQ9GyCmDg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-27T00:21:49.848370Z"},"content_sha256":"69a7ecefeb4b0c4931ab0f3350d5a86d347ae4da8502fbd7380d9d3d6db04d56","schema_version":"1.0","event_id":"sha256:69a7ecefeb4b0c4931ab0f3350d5a86d347ae4da8502fbd7380d9d3d6db04d56"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/HABCQY33QJ4HYN4JW3DOPY6EHZ/bundle.json","state_url":"https://pith.science/pith/HABCQY33QJ4HYN4JW3DOPY6EHZ/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/HABCQY33QJ4HYN4JW3DOPY6EHZ/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":"2026-05-27T00:21:49Z","links":{"resolver":"https://pith.science/pith/HABCQY33QJ4HYN4JW3DOPY6EHZ","bundle":"https://pith.science/pith/HABCQY33QJ4HYN4JW3DOPY6EHZ/bundle.json","state":"https://pith.science/pith/HABCQY33QJ4HYN4JW3DOPY6EHZ/state.json","well_known_bundle":"https://pith.science/.well-known/pith/HABCQY33QJ4HYN4JW3DOPY6EHZ/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2017:HABCQY33QJ4HYN4JW3DOPY6EHZ","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":"4125b3b89f07d84e7b85af5d6bfed66050e877417ce4e4fddc7dd6b279f5c881","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-07-16T01:32:17Z","title_canon_sha256":"2d8094d4a69b833214ceaa3c615c727343a6c5569aa22b4c4456db8a1fe8428b"},"schema_version":"1.0","source":{"id":"1707.04804","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1707.04804","created_at":"2026-05-18T00:40:12Z"},{"alias_kind":"arxiv_version","alias_value":"1707.04804v1","created_at":"2026-05-18T00:40:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1707.04804","created_at":"2026-05-18T00:40:12Z"},{"alias_kind":"pith_short_12","alias_value":"HABCQY33QJ4H","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_16","alias_value":"HABCQY33QJ4HYN4J","created_at":"2026-05-18T12:31:18Z"},{"alias_kind":"pith_short_8","alias_value":"HABCQY33","created_at":"2026-05-18T12:31:18Z"}],"graph_snapshots":[{"event_id":"sha256:69a7ecefeb4b0c4931ab0f3350d5a86d347ae4da8502fbd7380d9d3d6db04d56","target":"graph","created_at":"2026-05-18T00:40:12Z","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 completely determine the critical points of the normalized Eisenstein series $E_2(\\tau)$ of weight $2$. Although $E_2(\\tau)$ is not a modular form, our result shows that $E_2(\\tau)$ has at most one critical point in every fundamental domain of $\\Gamma_{0}(2)$. We also give a criteria for a fundamental domain containing a critical point of $E_2(\\tau)$. Furthermore, under the M\\\"obius transformation of $\\Gamma_{0}(2)$ action, all critical points can be mapped into the basic fundamental domain $F_0$ and their images are contained densely on three smooth curves. A geometric inter","authors_text":"Chang-shou Lin, Zhijie Chen","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-07-16T01:32:17Z","title":"Critical points of the classical Eisenstein series of weight two"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1707.04804","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":"sha256:749a3f9cf93be3b0c02df2d84a1ee2aa6aa7b7e6cad0eaba3b1cf3a1ac07d78e","target":"record","created_at":"2026-05-18T00:40:12Z","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":"4125b3b89f07d84e7b85af5d6bfed66050e877417ce4e4fddc7dd6b279f5c881","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-07-16T01:32:17Z","title_canon_sha256":"2d8094d4a69b833214ceaa3c615c727343a6c5569aa22b4c4456db8a1fe8428b"},"schema_version":"1.0","source":{"id":"1707.04804","kind":"arxiv","version":1}},"canonical_sha256":"380228637b82787c3789b6c6e7e3c43e4573bfaaf50793a598b9ffa52ef0ab38","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"380228637b82787c3789b6c6e7e3c43e4573bfaaf50793a598b9ffa52ef0ab38","first_computed_at":"2026-05-18T00:40:12.205719Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:40:12.205719Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ELWZiWr1ygN2nA9OBMocwTAJGIH0hcL+y3UUYuGTMpEURedK/1yEwGRQfkf2jmCJ0ByXSR+zfJsJA0iv3QQEDQ==","signature_status":"signed_v1","signed_at":"2026-05-18T00:40:12.206291Z","signed_message":"canonical_sha256_bytes"},"source_id":"1707.04804","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:749a3f9cf93be3b0c02df2d84a1ee2aa6aa7b7e6cad0eaba3b1cf3a1ac07d78e","sha256:69a7ecefeb4b0c4931ab0f3350d5a86d347ae4da8502fbd7380d9d3d6db04d56"],"state_sha256":"5d65141faf0ce29339f18acee149b6fad375f55447b28cd2926a7cd433856010"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"0q7DdnjVaaRXJEuEGG8FX9W4SE1Z6G89uNrqnKwzFaEEngF6CL52sFloli611E0Sr64T6hWXpMda/Y/nGS7iDg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-27T00:21:49.852359Z","bundle_sha256":"1d62a57a1e7e3c234b0e1eefa1260fb0d02d1efdcb4134d772d0d3513cdb3f9d"}}