{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2015:I7WARONN2PWRWY3P2YIGPZYPFA","short_pith_number":"pith:I7WARONN","canonical_record":{"source":{"id":"1508.07663","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-08-31T02:06:05Z","cross_cats_sorted":[],"title_canon_sha256":"2038fc66764ced5a9599176cc60fb0ad1754ce3423aef0cba31d6d6dc78ae56e","abstract_canon_sha256":"8a1fd8baa6618781405b9edd176cd0cf47c46f4d200e3ceb6118cbb472dc7c91"},"schema_version":"1.0"},"canonical_sha256":"47ec08b9add3ed1b636fd61067e70f2832a9b920305724549b07dd95fb8159b9","source":{"kind":"arxiv","id":"1508.07663","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.07663","created_at":"2026-07-05T03:49:06Z"},{"alias_kind":"arxiv_version","alias_value":"1508.07663v2","created_at":"2026-07-05T03:49:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.07663","created_at":"2026-07-05T03:49:06Z"},{"alias_kind":"pith_short_12","alias_value":"I7WARONN2PWR","created_at":"2026-07-05T03:49:06Z"},{"alias_kind":"pith_short_16","alias_value":"I7WARONN2PWRWY3P","created_at":"2026-07-05T03:49:06Z"},{"alias_kind":"pith_short_8","alias_value":"I7WARONN","created_at":"2026-07-05T03:49:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2015:I7WARONN2PWRWY3P2YIGPZYPFA","target":"record","payload":{"canonical_record":{"source":{"id":"1508.07663","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-08-31T02:06:05Z","cross_cats_sorted":[],"title_canon_sha256":"2038fc66764ced5a9599176cc60fb0ad1754ce3423aef0cba31d6d6dc78ae56e","abstract_canon_sha256":"8a1fd8baa6618781405b9edd176cd0cf47c46f4d200e3ceb6118cbb472dc7c91"},"schema_version":"1.0"},"canonical_sha256":"47ec08b9add3ed1b636fd61067e70f2832a9b920305724549b07dd95fb8159b9","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-07-05T03:49:06.352402Z","signature_b64":"dhuzNiNhfc0bPOgbwbj+RQkbxptm5XBtjOfIe1TEotSwoQE7ciC+sBz9B3u/U0Ebc7A+fIOwZKL2pBOhWQ3tBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"47ec08b9add3ed1b636fd61067e70f2832a9b920305724549b07dd95fb8159b9","last_reissued_at":"2026-07-05T03:49:06.351893Z","signature_status":"signed_v1","first_computed_at":"2026-07-05T03:49:06.351893Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1508.07663","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":"2026-07-05T03:49:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/fHxkjYT2p6zPDnLu+f4Z5QHWisjXptRuwMa/E6ZWxnB0rUCdk7ZDMZ+4d+7YKVSBgbz9dNw2Un06be7CAOCBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T10:20:01.352817Z"},"content_sha256":"39742b261c92cfef72fb6d0d890bebc7539d9d227e1d9262fed841ef2aabb2ab","schema_version":"1.0","event_id":"sha256:39742b261c92cfef72fb6d0d890bebc7539d9d227e1d9262fed841ef2aabb2ab"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2015:I7WARONN2PWRWY3P2YIGPZYPFA","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Possible indices for the Galois image of elliptic curves over Q","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"David Zywina","submitted_at":"2015-08-31T02:06:05Z","abstract_excerpt":"For a non-CM elliptic curve $E$ over the rationals, the Galois action on its torsion points can be expressed in terms of a Galois representation $\\rho_E : G \\to GL_2(\\hat{\\mathbb{Z}})$, where $G$ is the absolute Galois group of the rationals. A well-known theorem of Serre says that the image of $\\rho_E$ is open and hence has finite index in $GL_2(\\hat{\\mathbb{Z}})$. We will study what indices are possible assuming that we are willing to exclude a finite number of possible $j$-invariants from consideration. For example, we will show that there is a finite set $J$ of rational numbers such that i"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.07663","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1508.07663/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2026-07-05T03:49:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"RpRF1S+5Ujmn60aGTZwRNsHQ13UVkMke8F2b1gUPEn4yDdRdopmwN6tfUtBQ6BaSxJfIN9AaPJutNbOPGmytBw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-08-12T10:20:01.353450Z"},"content_sha256":"38fc0eef005f073e7c9ccf180fdfa0b435833999fff22aac27d096de985e5d92","schema_version":"1.0","event_id":"sha256:38fc0eef005f073e7c9ccf180fdfa0b435833999fff22aac27d096de985e5d92"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/I7WARONN2PWRWY3P2YIGPZYPFA/bundle.json","state_url":"https://pith.science/pith/I7WARONN2PWRWY3P2YIGPZYPFA/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/I7WARONN2PWRWY3P2YIGPZYPFA/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":"2026-08-12T10:20:01Z","links":{"resolver":"https://pith.science/pith/I7WARONN2PWRWY3P2YIGPZYPFA","bundle":"https://pith.science/pith/I7WARONN2PWRWY3P2YIGPZYPFA/bundle.json","state":"https://pith.science/pith/I7WARONN2PWRWY3P2YIGPZYPFA/state.json","well_known_bundle":"https://pith.science/.well-known/pith/I7WARONN2PWRWY3P2YIGPZYPFA/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:I7WARONN2PWRWY3P2YIGPZYPFA","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":"8a1fd8baa6618781405b9edd176cd0cf47c46f4d200e3ceb6118cbb472dc7c91","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-08-31T02:06:05Z","title_canon_sha256":"2038fc66764ced5a9599176cc60fb0ad1754ce3423aef0cba31d6d6dc78ae56e"},"schema_version":"1.0","source":{"id":"1508.07663","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1508.07663","created_at":"2026-07-05T03:49:06Z"},{"alias_kind":"arxiv_version","alias_value":"1508.07663v2","created_at":"2026-07-05T03:49:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1508.07663","created_at":"2026-07-05T03:49:06Z"},{"alias_kind":"pith_short_12","alias_value":"I7WARONN2PWR","created_at":"2026-07-05T03:49:06Z"},{"alias_kind":"pith_short_16","alias_value":"I7WARONN2PWRWY3P","created_at":"2026-07-05T03:49:06Z"},{"alias_kind":"pith_short_8","alias_value":"I7WARONN","created_at":"2026-07-05T03:49:06Z"}],"graph_snapshots":[{"event_id":"sha256:38fc0eef005f073e7c9ccf180fdfa0b435833999fff22aac27d096de985e5d92","target":"graph","created_at":"2026-07-05T03:49:06Z","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"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/1508.07663/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"For a non-CM elliptic curve $E$ over the rationals, the Galois action on its torsion points can be expressed in terms of a Galois representation $\\rho_E : G \\to GL_2(\\hat{\\mathbb{Z}})$, where $G$ is the absolute Galois group of the rationals. A well-known theorem of Serre says that the image of $\\rho_E$ is open and hence has finite index in $GL_2(\\hat{\\mathbb{Z}})$. We will study what indices are possible assuming that we are willing to exclude a finite number of possible $j$-invariants from consideration. For example, we will show that there is a finite set $J$ of rational numbers such that i","authors_text":"David Zywina","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-08-31T02:06:05Z","title":"Possible indices for the Galois image of elliptic curves over Q"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1508.07663","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":"sha256:39742b261c92cfef72fb6d0d890bebc7539d9d227e1d9262fed841ef2aabb2ab","target":"record","created_at":"2026-07-05T03:49:06Z","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":"8a1fd8baa6618781405b9edd176cd0cf47c46f4d200e3ceb6118cbb472dc7c91","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-08-31T02:06:05Z","title_canon_sha256":"2038fc66764ced5a9599176cc60fb0ad1754ce3423aef0cba31d6d6dc78ae56e"},"schema_version":"1.0","source":{"id":"1508.07663","kind":"arxiv","version":2}},"canonical_sha256":"47ec08b9add3ed1b636fd61067e70f2832a9b920305724549b07dd95fb8159b9","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"47ec08b9add3ed1b636fd61067e70f2832a9b920305724549b07dd95fb8159b9","first_computed_at":"2026-07-05T03:49:06.351893Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T03:49:06.351893Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"dhuzNiNhfc0bPOgbwbj+RQkbxptm5XBtjOfIe1TEotSwoQE7ciC+sBz9B3u/U0Ebc7A+fIOwZKL2pBOhWQ3tBA==","signature_status":"signed_v1","signed_at":"2026-07-05T03:49:06.352402Z","signed_message":"canonical_sha256_bytes"},"source_id":"1508.07663","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:39742b261c92cfef72fb6d0d890bebc7539d9d227e1d9262fed841ef2aabb2ab","sha256:38fc0eef005f073e7c9ccf180fdfa0b435833999fff22aac27d096de985e5d92"],"state_sha256":"6564fb0939ecf83c2e5bfcdb900235e8bd7955898174f320baf770fff7f4825a"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"H6Yb/7TBSFoOcEaDSDd+uJuFjqDaECDBcLaiCw/l4hCUX3xHmTSRwsVXLx2E/H/TTnJwDXoQ5P2eH9pDYYehAg==","signed_message":"bundle_sha256_bytes","signed_at":"2026-08-12T10:20:01.358420Z","bundle_sha256":"16f47bee8e72d80650a9c71968f14f5687a855e406f7fe612a156a11998b4fa1"}}