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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."},{"attestation":"unclaimed","claim_id":"C2","kind":"weakest_assumption","source":"verdict.weakest_assumption","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C3","kind":"one_line_summary","source":"verdict.one_line_summary","status":"machine_extracted","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."},{"attestation":"unclaimed","claim_id":"C4","kind":"headline","source":"verdict.pith_extraction.headline","status":"machine_extracted","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."}],"snapshot_sha256":"98ea61f461e8effa35c48bfd740ba4645076c01e1c8bbfaf892e1e9a99844275"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2604.11330/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"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","authors_text":"Alexandru Ghitza, Dhruv Gupta, Maximilian Kortge","cross_cats":[],"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.","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-04-13T11:32:10Z","title":"The solvability of the inverse volcano problem over non-prime finite fields"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2604.11330","kind":"arxiv","version":2},"verdict":{"created_at":"2026-05-10T15:24:10.536781Z","id":"bc118608-3d9b-449f-ba9a-ae56e18c4fb9","model_set":{"reader":"grok-4.3"},"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","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.","strongest_claim":"We generalise the results of Bambury-Campagna-Pazuki by providing a precise framework for the inverse volcano problem over F_{p^k}. 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