The solvability of the inverse volcano problem over non-prime finite fields
Pith reviewed 2026-05-10 15:24 UTC · model grok-4.3
The pith
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.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
The paper establishes a precise framework for the inverse volcano problem over F_{p^k}: for an ℓ-volcano V of depth d, solvability over F_{p^k} is determined by the relation between d and r = v_ℓ(k). When r is small relative to d, there exist infinitely many primes p such that V occurs as a connected component of the ℓ-isogeny graph over F_{p^k}. When r is large relative to d, the problem is unsolvable for many V; in the remaining cases solvability holds conditionally on a variant of the Cohen-Lenstra heuristics for class groups of imaginary quadratic fields, with computational evidence provided in support.
What carries the argument
The ℓ-volcano graph of depth d arising as a connected component in the graph of ℓ-isogenies of ordinary elliptic curves over F_{p^k}, whose realization is controlled by comparison of d to the ℓ-valuation r of the extension degree k.
If this is right
- When r is smaller than d there are infinitely many primes p for which the given volcano V appears as a component over F_{p^k}.
- When r is larger than d the inverse problem is unsolvable for most choices of V.
- In the remaining cases where r is large, solvability is conditional on the variant Cohen-Lenstra heuristic.
- Computational checks confirm the heuristic predictions in the examined range.
Where Pith is reading between the lines
- The framework restricts which volcano shapes can appear for a fixed extension degree, limiting the possible connected components in isogeny graphs over F_{p^k}.
- Removing the heuristic would require proving the relevant Cohen-Lenstra variant for the class groups that arise in the large-r regime.
- The dependence on r suggests that choosing k divisible by high powers of ℓ can forbid certain volcano depths from occurring.
Load-bearing premise
A variant of the Cohen-Lenstra heuristics for class groups of imaginary quadratic fields holds when the ℓ-valuation r of k is large compared to the volcano depth d.
What would settle it
An explicit volcano V with r large relative to d for which the inverse problem is solvable, together with a class group whose distribution deviates from the modified Cohen-Lenstra prediction, would disprove the conditional solvability statements.
read the original abstract
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 $\mathbf{F}_{p^2}$. In this paper, we generalise the results of Bambury-Campagna-Pazuki by providing a precise framework for the inverse volcano problem over $\mathbf{F}_{p^k}$. The solvability of the problem for an $\ell$-volcano graph $V$ of depth $d$ is typically determined by the relation between $d$ and the $\ell$-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 $V$. The situation where $r$ is large in comparison to $d$ is more delicate: in many cases we prove that the inverse problem for $V$ is unsolvable; in a few other cases the problem appears to be solvable, but our proof of this is conditional on a variant of the Cohen-Lenstra heuristics for class groups of imaginary quadratic fields. We provide some computational evidence in support of these modified heuristics.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript generalizes the inverse volcano problem to F_{p^k} for k>1. It provides a framework classifying solvability of realizing a given ℓ-volcano V of depth d over F_{p^k} in terms of the relation between d and r = v_ℓ(k). When r is small relative to d, it proves unconditionally that infinitely many primes p exist realizing V. When r is large relative to d, it proves unsolvability in many cases and asserts solvability in a few others conditionally on a variant of the Cohen-Lenstra heuristics for class groups of imaginary quadratic fields, supported by computational evidence.
Significance. If the results hold, the work supplies a precise extension of Bambury-Campagna-Pazuki to non-prime fields, distinguishing regimes by the ℓ-valuation of the extension degree. The unconditional theorems for small r rest on standard tools (Kohel's volcano structure and class-number formulas) and are load-bearing contributions. The conditional results for large r are transparently flagged and include computational checks, which is appropriate given the prevalence of such heuristics in arithmetic statistics.
major comments (2)
- [§4] §4 (large-r regime): The positive solvability claims for certain volcanoes when r ≳ d (e.g., the statements following Conjecture 4.3) rest on an unproven variant of the Cohen-Lenstra heuristic whose only support is computational evidence for small cases. This is load-bearing for the 'appears to be solvable' assertions in the delicate regime the authors themselves flag.
- [Theorem 5.4] Theorem 5.4 and surrounding discussion: The separation between unconditional unsolvability proofs and the heuristic-dependent solvability cases is not fully explicit; it is unclear whether the class-group conditions used in the unsolvability arguments remain independent of the modified heuristic invoked elsewhere.
minor comments (2)
- [Introduction] The introduction would benefit from an explicit table or diagram summarizing the solvability status for each (d,r) regime, including which cases are unconditional, which are unsolvable, and which are conditional.
- [§2] Notation for the ℓ-valuation r of k is introduced after the main statements; moving the definition to §2 would improve readability.
Simulated Author's Rebuttal
Thank you for the referee's careful review and constructive feedback on our manuscript generalizing the inverse volcano problem to F_{p^k}. We address each major comment point by point below, indicating where revisions will be made to improve clarity.
read point-by-point responses
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Referee: [§4] §4 (large-r regime): The positive solvability claims for certain volcanoes when r ≳ d (e.g., the statements following Conjecture 4.3) rest on an unproven variant of the Cohen-Lenstra heuristic whose only support is computational evidence for small cases. This is load-bearing for the 'appears to be solvable' assertions in the delicate regime the authors themselves flag.
Authors: We agree that the solvability assertions in §4 for the large-r regime (following Conjecture 4.3) are conditional on the unproven variant of the Cohen-Lenstra heuristic and rest on computational evidence for small cases. The manuscript already describes these as conditional and provides the supporting computations. To address the concern that these claims are load-bearing, we will revise the text in §4 to more prominently label the relevant statements as 'conditionally solvable under Conjecture 4.3' and add an explicit caveat that the heuristic remains unproven. This will make the conjectural status unambiguous without altering the conditional nature of the results. revision: yes
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Referee: [Theorem 5.4] Theorem 5.4 and surrounding discussion: The separation between unconditional unsolvability proofs and the heuristic-dependent solvability cases is not fully explicit; it is unclear whether the class-group conditions used in the unsolvability arguments remain independent of the modified heuristic invoked elsewhere.
Authors: We thank the referee for highlighting this point. The unsolvability results in Theorem 5.4 rely on explicit, provable conditions on class groups derived from Kohel's volcano structure and class number formulas; these are independent of the modified Cohen-Lenstra variant used only for the positive solvability cases in the large-r regime. To make the separation fully explicit, we will insert a clarifying remark or short paragraph in the discussion following Theorem 5.4 that distinguishes the two sets of class-group conditions and notes their independence. This revision will resolve the ambiguity. revision: yes
- Unconditional proof of solvability for the inverse volcano problem in the large-r regime cases that depend on the variant of the Cohen-Lenstra heuristic (as this remains an open conjecture).
Circularity Check
No circularity: unconditional claims rest on external theorems; conditional claims are explicitly flagged
full rationale
The paper's core unconditional result (infinitely many p when r small relative to d) is derived from Kohel's volcano theorem and standard class-number formulas for imaginary quadratic fields, both independent of the target solvability statement. Cases with r large relative to d are either proven unsolvable outright or stated as conditional on a variant of the Cohen-Lenstra heuristics, with separate computational checks supplied rather than any parameter fitting or self-referential derivation. No self-citations are load-bearing, no ansatz is smuggled, and no prediction reduces to a fitted input by construction. The derivation chain is self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (3)
- standard math Kohel's theorem that the connected components of the ℓ-isogeny graph of ordinary elliptic curves are volcano graphs
- standard math The correspondence between ordinary elliptic curves over F_{p^k} and ideal classes in orders of imaginary quadratic fields
- standard math Existence and basic properties of the class group of an imaginary quadratic order
discussion (0)
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