{"paper":{"title":"The solvability of the inverse volcano problem over non-prime finite fields","license":"http://creativecommons.org/licenses/by/4.0/","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.","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alexandru Ghitza, Dhruv Gupta, Maximilian Kortge","submitted_at":"2026-04-13T11:32:10Z","abstract_excerpt":"For a finite field $\\mathbf{F}_{p^k}$ and a prime $\\ell \\neq p$, consider the graph $G$ of $\\ell$-isogenies between ordinary elliptic curves over $\\mathbf{F}_{p^k}$. Kohel proved that the connected components of $G$ have a remarkable structure, now called an $\\ell$-volcano graph. Bambury, Campagna, and Pazuki investigated the inverse volcano problem: given a volcano graph $V$, can one find it as a connected component of $G$ over $\\mathbf{F}_{p^k}$? They gave a complete positive answer over $\\mathbf{F}_p$, and described a specific counterexample over $\\mathbf{F}_{p^2}$.\n  In this paper, we gene"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We generalise the results of Bambury-Campagna-Pazuki by providing a precise framework for the inverse volcano problem over F_{p^k}. 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.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"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.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"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.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"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.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"98ea61f461e8effa35c48bfd740ba4645076c01e1c8bbfaf892e1e9a99844275"},"source":{"id":"2604.11330","kind":"arxiv","version":2},"verdict":{"id":"bc118608-3d9b-449f-ba9a-ae56e18c4fb9","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T15:24:10.536781Z","strongest_claim":"We generalise the results of Bambury-Campagna-Pazuki by providing a precise framework for the inverse volcano problem over F_{p^k}. 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"}