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

This paper generalizes previous results on the inverse volcano problem by establishing a precise framework over non-prime finite fields $\mathbf{F}_{p^k}$ , demonstrating that the solvability of realizing a given $\ell$ -volcano graph depends critically on the relationship between the graph's depth and the $\ell$ -valuation of the field extension degree, with unconditional existence proofs for small valuations and conditional unsolvability or solvability results for large valuations based on modified Cohen-Lenstra heuristics.

Original authors: Alexandru Ghitza, Dhruv Gupta, Maximilian Kortge

Published 2026-04-14

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CC BY 4.0

Original authors: Alexandru Ghitza, Dhruv Gupta, Maximilian Kortge

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

The Big Picture: The Volcano Hunt

Imagine you are a geologist, but instead of studying real volcanoes, you are studying mathematical volcanoes. These aren't made of rock and lava; they are made of numbers, graphs, and shapes called "elliptic curves."

In the world of cryptography (the math behind secure internet connections), these mathematical volcanoes are very important. They represent how different types of these curves are connected to each other.

The Crater: The top of the volcano. It's a ring of vertices (points) where the curves are "ordinary" and stable.
The Lava Flows: Trees of vertices hanging down from the crater. As you go deeper, the curves change, and eventually, the lava stops at a specific depth.

The Problem: The "Inverse" Volcano

Usually, mathematicians start with a specific field of numbers (like a specific type of soil) and ask: "What kind of volcano grows here?"

This paper asks the Inverse Volcano Problem:
"If I want to see a specific volcano (with a specific crater shape and a specific depth of lava), can I find a field of numbers where it grows?"

Think of it like this: You have a blueprint for a very specific, weird-looking volcano. You want to know if there is a patch of land (a mathematical field) where this exact volcano can be built.

The Twist: Prime vs. Non-Prime Fields

In a previous study, mathematicians found that if you look at "prime" fields (simple, single-number fields), the answer is almost always YES. You can almost always find a field where your volcano grows.

But this paper tackles the harder case: Non-prime fields (fields made of combinations of numbers, like $p^k$ where $k > 1$ ).

The Bad News: Sometimes, the answer is NO. No matter how hard you look, that specific volcano cannot grow in these complex fields.
The Good News: The authors figured out exactly when it's possible and when it's impossible. They created a massive "Rulebook" (a table in the paper) that tells you, based on the volcano's shape and the field's properties, whether the volcano will appear.

The Secret Ingredient: The "Class Group"

How do they solve this? They don't just look at the volcano; they look at the soil's DNA.

In mathematics, every field has a hidden structure called a Class Group. You can think of this as the "genetic code" of the soil.

The shape of the volcano's crater depends on how the "prime number" $\ell$ behaves in this genetic code.
The depth of the lava depends on how many "generations" of this code you can stretch out.

The authors discovered that for a volcano to exist, the soil's genetic code must have a very specific "vibration" or order. If the code is too "tight" (divisible by the prime number too many times), the volcano collapses and cannot form.

The "Explosive" Primes

The paper talks about "k-explosive primes."

Imagine you are looking for a specific volcano.
A "prime" is a candidate field.
If that field allows your volcano to grow, it "explodes" into existence.
The paper asks: Are there infinitely many fields that make this volcano explode?

The Results:

Sometimes, Yes: For many volcano shapes, there are infinitely many fields where they grow.
Sometimes, No: For certain shapes (like a 2-volcano with a specific crater), the paper proves they never grow in these complex fields. It's like trying to grow a cactus in a swamp; the environment just doesn't support it.
The "Maybe" Zone: For some very complex shapes, the answer depends on a famous mathematical guess called the Cohen-Lenstra Heuristics. This is like a weather forecast for number theory. The authors say, "If our weather forecast is right, then yes, these volcanoes exist. If not, we don't know yet."

The "Lava" Analogy for Depth

Imagine the volcano has a depth $d$ .

Depth 0: Just the crater. (Easy to find).
Depth 1, 2, 3...: The lava flows get longer.
The paper finds that if the "field" (the soil) is too "rich" in a specific prime number (mathematicians call this the $\ell$ -adic valuation, or how many times the prime divides the field size), the lava flow gets cut off. The volcano gets "stunted."

Summary of the Discovery

The authors, Alexandru Ghitza, Dhruv Gupta, and Maximilian Kortge, have built a comprehensive map.

If you have a simple volcano: It grows almost everywhere.
If you have a complex, deep volcano: It only grows if the soil's "genetic code" (the class group) has a specific structure.
If the soil is "too rich" in the wrong way: The volcano is impossible.

They didn't just guess; they used deep algebra (looking at how numbers divide and multiply) to prove exactly which combinations work and which don't. This helps cryptographers understand the limits of the mathematical structures they use to secure our data.

In a nutshell: They figured out the exact recipe for growing mathematical volcanoes in complex fields, proving that some shapes are impossible to grow, while others are guaranteed to appear infinitely often.

1. Problem Statement

The paper addresses the Inverse Volcano Problem in the context of arithmetic geometry and cryptography.

Context: The graph $G_\ell(\mathbb{F}_{p^k})$ of $\ell$ -isogenies between ordinary elliptic curves over a finite field $\mathbb{F}_{p^k}$ decomposes into connected components known as $\ell$ -volcanoes. These are graphs resembling a volcano, consisting of a "crater" (a regular graph) and "lava flows" (trees) descending to a finite depth $d$ .
The Problem: Given a fixed prime $\ell$ , a specific volcano structure $V$ (defined by its crater type $V_0$ , depth $d$ , and the prime $\ell$ ), and an integer $k \geq 1$ , does there exist a prime $p \neq \ell$ such that $V$ appears as a connected component in $G_\ell(\mathbb{F}_{p^k})$ ? If so, how many such primes exist?
Previous Work: Bambury–Campagna–Pazuki [1] solved this for $k=1$ (prime fields), proving that any volcano appears for infinitely many primes. However, for $k > 1$ (non-prime fields), the situation is delicate; specific counterexamples exist where no such primes exist (e.g., a specific 2-volcano of depth 1 never appears in $\mathbb{F}_{p^2}$ ).

2. Methodology

The authors develop a precise algebraic framework to determine the existence of such primes by linking the geometry of the isogeny graph to the arithmetic of imaginary quadratic fields.

A. Algebraic Characterization

The core of the methodology relies on the correspondence between elliptic curves with Complex Multiplication (CM) and orders in imaginary quadratic fields.

Volcano Structure: A volcano of depth $d$ corresponds to a chain of orders $\mathcal{O}_0 \supset \mathcal{O}_1 \supset \dots \supset \mathcal{O}_d$ in an imaginary quadratic field $K$ , where $[\mathcal{O}_i : \mathcal{O}_{i+1}] = \ell$ . The crater $\mathcal{O}_0$ determines the graph's topology (type $I_1, R_1, R_2, S_n$ , etc.) based on the splitting behavior of $\ell$ in $\mathcal{O}_0$ and the order of prime ideals in the class group $\text{Cl}(\mathcal{O}_0)$ .
Existence Criterion (Theorem 4.2): The existence of a volcano $V$ $V$ in $G_\ell(\mathbb{F}_{p^k})$ $G_{ℓ} (F_{p^{k}})$ is equivalent to the existence of a prime $p$ $p$ such that the Frobenius element $\sigma_p$ $σ_{p}$ in the Galois group of the ring class field $H_{d+1}/K$ $H_{d + 1} / K$ satisfies specific order conditions:
- $\text{ord}(\sigma_p \text{ in } \text{Gal}(H_d/K)) \mid k$
- $\text{ord}(\sigma_p \text{ in } \text{Gal}(H_{d+1}/K)) \nmid k$
Group Theoretic Reduction: Using the Chebotarev Density Theorem, the problem reduces to checking if a specific set $X$ of elements in the Galois group (isomorphic to the class group $\text{Cl}(\mathcal{O}_{d+1})$ ) is non-empty. This is further reduced to analyzing the intersection of the kernel of the natural map between class groups and the subgroup of elements divisible by $\ell^r$ (where $r = \nu_\ell(k)$ ).

B. Analysis of Class Groups

The authors perform a deep analysis of the structure of the Sylow $\ell$ -subgroups of the class groups of orders $\mathcal{O}_d$ as the conductor increases.

Kernels ( $\kappa_d$ ): They explicitly describe the kernels $\kappa_d$ of the maps $\text{Cl}(\mathcal{O}_d) \to \text{Cl}(\mathcal{O}_0)$ for various ramification types (Split, Inert, Ramified) of $\ell$ .
Splitting Conditions: They investigate when the short exact sequence $1 \to \kappa_d \to S_d \to S_0 \to 1$ splits, which is crucial for determining the size of the set $X$ .
2-Torsion: Special attention is paid to the case $\ell=2$ , utilizing genus theory to describe 2-torsion in class groups and determining conditions under which the sequences split.

C. Heuristics and Computation

For cases where unconditional proofs are difficult (specifically when the rank of the class group changes with depth), the authors:

Modify Cohen–Lenstra Heuristics: They propose a modification to the Cohen–Lenstra heuristics to predict the distribution of class groups of orders $\mathcal{O}_d$ relative to $\mathcal{O}_0$ . This allows them to estimate the probability that the necessary algebraic conditions are met.
Computational Verification: They use high-performance computing (University of Melbourne's "Spartan") to verify these heuristics for discriminants up to $2^{37}$ , providing strong empirical evidence for their conditional results.

3. Key Contributions and Results

The paper provides a complete classification of the solvability of the inverse volcano problem for $k > 1$ . The results are summarized in a comprehensive table (Table 1 in the paper) based on the crater type $V_0$ , the depth $d$ , and the $\ell$ -adic valuation of $k$ (denoted $r$ ).

A. Unconditional Results

The authors prove that for many configurations, the set of $k$ -explosive primes is either infinite (positive density) or empty.

Depth Zero ( $d=0$ ): They generalize previous results, showing that for any crater type, there are infinitely many $k$ -explosive primes unless specific divisibility conditions on $k$ and $\ell$ are met (e.g., if $\ell$ splits and $\ell-1 \mid k$ ).
Depth $d \geq 1$ :
- Infinite Existence: For most crater types ( $S_n, I_1, R_2, R_1$ ) and depths, if the valuation $r$ is sufficiently small (specifically $r < d$ or $r < d+1$ depending on the type), there are infinitely many $k$ -explosive primes.
- Non-Existence: They prove that for certain configurations, no such primes exist.
  - Example: If $V_0 = R_1$ (principally ramified) and $r \geq d+1$ (for $\ell \neq 3$ ), no $k$ -explosive primes exist.
  - Example: If $V_0 = S_n$ (split) and $\ell=2$ , specific constraints on $r$ and $d$ lead to non-existence.
- Refined Counterexamples: They generalize the counterexample from [1] (Proposition 1.2), proving that the 2-volcano of depth 1 never appears in $\mathbb{F}_{p^2}$ (Theorem 5.10).

B. Conditional Results

For cases where the behavior of the class group ranks is not fully determined (marked with ?? or * in the summary table), the authors provide conditional results:

Modified Heuristics: Assuming their modified Cohen–Lenstra heuristics hold, they prove that for cases like $V_0 = I_1$ (inert) with $r \geq d$ , there are infinitely many $k$ -explosive primes.
Computational Evidence: The paper presents graphs (Figure 2) showing that the proportion of imaginary quadratic fields satisfying the required class group properties converges to the predicted probability as the discriminant bound increases, strongly supporting the conditional claims.

4. Significance

Theoretical Completeness: This work provides the first comprehensive framework for the inverse volcano problem over non-prime finite fields ( $k > 1$ ), moving beyond the $k=1$ case. It establishes a dichotomy: for a fixed volcano, the set of primes is either empty or has positive density.
Cryptographic Implications: The structure of isogeny graphs is fundamental to isogeny-based cryptography (e.g., CSIDH, SIKE). Understanding which volcano structures can appear over specific field extensions is crucial for analyzing the security and properties of cryptographic schemes defined over these fields.
Arithmetic Geometry: The paper advances the understanding of the relationship between isogeny graphs and the class groups of orders in imaginary quadratic fields, particularly regarding the behavior of $\ell$ -torsion as the conductor grows.
Heuristic Framework: The modification of the Cohen–Lenstra heuristics to include nested orders provides a new tool for predicting the distribution of class groups in families of orders, a problem of independent interest in number theory.

In summary, the paper successfully characterizes the solvability of the inverse volcano problem, providing a mix of unconditional theorems and strongly supported conditional results, thereby filling a significant gap in the arithmetic theory of elliptic curve isogeny graphs.