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The solvability of the problem for an ℓ-volcano graph V of depth d is typically determined by the relation between d and the ℓ-valuation r of k. When r is small in comparison to d, we prove that there are infinitely many primes p solving the inverse problem for V.","one_line_summary":"The inverse ℓ-volcano problem over F_{p^k} is solvable for infinitely many p when d exceeds r, often unsolvable when r exceeds d, and conditionally solvable in remaining cases under a variant of the Cohen-Lenstra heuristics.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"A variant of the Cohen-Lenstra heuristics for class groups of imaginary quadratic fields, invoked to handle the cases in which r is large compared to d.","pith_extraction_headline":"Whether a given ℓ-volcano of depth d appears in the isogeny graph over F_{p^k} depends on how d compares to the ℓ-valuation r of k."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.11330/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":"2604.11330","created_at":"2026-06-01T01:02:39.539080+00:00"},{"alias_kind":"arxiv_version","alias_value":"2604.11330v2","created_at":"2026-06-01T01:02:39.539080+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.11330","created_at":"2026-06-01T01:02:39.539080+00:00"},{"alias_kind":"pith_short_12","alias_value":"JF5UQQLWY4BT","created_at":"2026-06-01T01:02:39.539080+00:00"},{"alias_kind":"pith_short_16","alias_value":"JF5UQQLWY4BT6I3H","created_at":"2026-06-01T01:02:39.539080+00:00"},{"alias_kind":"pith_short_8","alias_value":"JF5UQQLW","created_at":"2026-06-01T01:02:39.539080+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/JF5UQQLWY4BT6I3HLIMA7AZIQE","json":"https://pith.science/pith/JF5UQQLWY4BT6I3HLIMA7AZIQE.json","graph_json":"https://pith.science/api/pith-number/JF5UQQLWY4BT6I3HLIMA7AZIQE/graph.json","events_json":"https://pith.science/api/pith-number/JF5UQQLWY4BT6I3HLIMA7AZIQE/events.json","paper":"https://pith.science/paper/JF5UQQLW"},"agent_actions":{"view_html":"https://pith.science/pith/JF5UQQLWY4BT6I3HLIMA7AZIQE","download_json":"https://pith.science/pith/JF5UQQLWY4BT6I3HLIMA7AZIQE.json","view_paper":"https://pith.science/paper/JF5UQQLW","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2604.11330&json=true","fetch_graph":"https://pith.science/api/pith-number/JF5UQQLWY4BT6I3HLIMA7AZIQE/graph.json","fetch_events":"https://pith.science/api/pith-number/JF5UQQLWY4BT6I3HLIMA7AZIQE/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/JF5UQQLWY4BT6I3HLIMA7AZIQE/action/timestamp_anchor","attest_storage":"https://pith.science/pith/JF5UQQLWY4BT6I3HLIMA7AZIQE/action/storage_attestation","attest_author":"https://pith.science/pith/JF5UQQLWY4BT6I3HLIMA7AZIQE/action/author_attestation","sign_citation":"https://pith.science/pith/JF5UQQLWY4BT6I3HLIMA7AZIQE/action/citation_signature","submit_replication":"https://pith.science/pith/JF5UQQLWY4BT6I3HLIMA7AZIQE/action/replication_record"}},"created_at":"2026-06-01T01:02:39.539080+00:00","updated_at":"2026-06-01T01:02:39.539080+00:00"}