Pith Number

pith:JF5UQQLW

pith:2026:JF5UQQLWY4BT6I3HLIMA7AZIQE

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The solvability of the inverse volcano problem over non-prime finite fields

Alexandru Ghitza, Dhruv Gupta, Maximilian Kortge

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.

arxiv:2604.11330 v2 · 2026-04-13 · math.NT

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Claims

C1strongest 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.

C2weakest 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.

C3one 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.

Receipt and verification

First computed	2026-06-01T01:02:39.538956Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

497b484176c7033f23675a180f832881041b6eaf321f9c7f9953d56aa0d1b5cc

Aliases

arxiv: 2604.11330 · arxiv_version: 2604.11330v2 · doi: 10.48550/arxiv.2604.11330 · pith_short_12: JF5UQQLWY4BT · pith_short_16: JF5UQQLWY4BT6I3H · pith_short_8: JF5UQQLW

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/JF5UQQLWY4BT6I3HLIMA7AZIQE \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: 497b484176c7033f23675a180f832881041b6eaf321f9c7f9953d56aa0d1b5cc

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "9425655c4a71bcdd0f5819900e2b56905929810d2f289fc79e354e9cf4f76697",
    "cross_cats_sorted": [],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math.NT",
    "submitted_at": "2026-04-13T11:32:10Z",
    "title_canon_sha256": "05d9c0fee245c7ab0cdfafedfe249f2b69a4575cb122b78aeda5c2e014bbe97c"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2604.11330",
    "kind": "arxiv",
    "version": 2
  }
}