{"record_type":"pith_number_record","schema_url":"https://pith.science/schemas/pith-number/v1.json","pith_number":"pith:2026:JEL2WLA5NW3I3ULTJNR2CUBKCY","short_pith_number":"pith:JEL2WLA5","schema_version":"1.0","canonical_sha256":"4917ab2c1d6db68dd1734b63a1502a1619ba4b99f8d78733e6b1f9e04d47bef5","source":{"kind":"arxiv","id":"2608.02494","version":1},"attestation_state":"computed","paper":{"title":"Isogeny graphs of elliptic curves in characteristic zero","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.CO"],"primary_cat":"math.NT","authors_text":"Alexander J. Barrios, Enrique Gonz\\'alez-Jim\\'enez, Ivan Novak","submitted_at":"2026-08-03T16:59:45Z","abstract_excerpt":"For an elliptic curve $E$ defined over a field $K$ of characteristic $0$ with $\\operatorname{End}_K \\! E \\cong \\mathbb{Z}$, we classify which isogeny graphs $\\mathcal{G}(E/K)$ can occur. We first show that $\\mathcal{G}(E/K)$ decomposes as a weak Cartesian product of its $p$-primary isogeny graphs, one for each prime $p$, thereby reducing the problem to classifying $p$-primary isogeny graphs. We then show that each such graph is isomorphic, as an edge-weighted graph, to a member of an explicit family of edge-weighted graphs $\\mathcal{H}_{p^k}^r$ and $\\mathcal{H}_{p^{\\infty,+}}^r$, every member "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2608.02494","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2026-08-03T16:59:45Z","cross_cats_sorted":["math.AG","math.CO"],"title_canon_sha256":"3e9def984c91c00f7ee5b245c107f99e07fa658900a3d9f6f92a20622ce0bce2","abstract_canon_sha256":"82ec014ab40283ad389a6aab540abc8c1a88f7fd9a283ce2505b01ee2d70bec4"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-08-04T02:43:47.760577Z","signature_b64":"uYm1n3L8GAH8YyncvT6lk1AvwlUTMTNxXO8VxjMp9c8/etp1NZy4XDcVpRpaXKUQB/diNRhLCrqRogfKYiaIBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"4917ab2c1d6db68dd1734b63a1502a1619ba4b99f8d78733e6b1f9e04d47bef5","last_reissued_at":"2026-08-04T02:43:47.758140Z","signature_status":"signed_v1","first_computed_at":"2026-08-04T02:43:47.758140Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Isogeny graphs of elliptic curves in characteristic zero","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AG","math.CO"],"primary_cat":"math.NT","authors_text":"Alexander J. Barrios, Enrique Gonz\\'alez-Jim\\'enez, Ivan Novak","submitted_at":"2026-08-03T16:59:45Z","abstract_excerpt":"For an elliptic curve $E$ defined over a field $K$ of characteristic $0$ with $\\operatorname{End}_K \\! E \\cong \\mathbb{Z}$, we classify which isogeny graphs $\\mathcal{G}(E/K)$ can occur. We first show that $\\mathcal{G}(E/K)$ decomposes as a weak Cartesian product of its $p$-primary isogeny graphs, one for each prime $p$, thereby reducing the problem to classifying $p$-primary isogeny graphs. We then show that each such graph is isomorphic, as an edge-weighted graph, to a member of an explicit family of edge-weighted graphs $\\mathcal{H}_{p^k}^r$ and $\\mathcal{H}_{p^{\\infty,+}}^r$, every member "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2608.02494","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2608.02494/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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2608.02494","created_at":"2026-08-04T02:43:47.760576+00:00"},{"alias_kind":"arxiv_version","alias_value":"2608.02494v1","created_at":"2026-08-04T02:43:47.760576+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2608.02494","created_at":"2026-08-04T02:43:47.760576+00:00"},{"alias_kind":"pith_short_12","alias_value":"JEL2WLA5NW3I","created_at":"2026-08-04T02:43:47.760576+00:00"},{"alias_kind":"pith_short_16","alias_value":"JEL2WLA5NW3I3ULT","created_at":"2026-08-04T02:43:47.760576+00:00"},{"alias_kind":"pith_short_8","alias_value":"JEL2WLA5","created_at":"2026-08-04T02:43:47.760576+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/JEL2WLA5NW3I3ULTJNR2CUBKCY","json":"https://pith.science/pith/JEL2WLA5NW3I3ULTJNR2CUBKCY.json","graph_json":"https://pith.science/api/pith-number/JEL2WLA5NW3I3ULTJNR2CUBKCY/graph.json","events_json":"https://pith.science/api/pith-number/JEL2WLA5NW3I3ULTJNR2CUBKCY/events.json","paper":"https://pith.science/paper/JEL2WLA5"},"agent_actions":{"view_html":"https://pith.science/pith/JEL2WLA5NW3I3ULTJNR2CUBKCY","download_json":"https://pith.science/pith/JEL2WLA5NW3I3ULTJNR2CUBKCY.json","view_paper":"https://pith.science/paper/JEL2WLA5","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2608.02494&json=true","fetch_graph":"https://pith.science/api/pith-number/JEL2WLA5NW3I3ULTJNR2CUBKCY/graph.json","fetch_events":"https://pith.science/api/pith-number/JEL2WLA5NW3I3ULTJNR2CUBKCY/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JEL2WLA5NW3I3ULTJNR2CUBKCY/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JEL2WLA5NW3I3ULTJNR2CUBKCY/action/storage_attestation","attest_author":"https://pith.science/pith/JEL2WLA5NW3I3ULTJNR2CUBKCY/action/author_attestation","sign_citation":"https://pith.science/pith/JEL2WLA5NW3I3ULTJNR2CUBKCY/action/citation_signature","submit_replication":"https://pith.science/pith/JEL2WLA5NW3I3ULTJNR2CUBKCY/action/replication_record"}},"created_at":"2026-08-04T02:43:47.760576+00:00","updated_at":"2026-08-04T02:43:47.760576+00:00"}