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Distributions of Iwasawa $\lambda$ -invariants of $\mathbf{Z}_p$ -towers over supersingular isogeny graphs

This paper investigates the distribution of Iwasawa $\lambda$ -invariants for constant $\mathbf{Z}_p$ -towers over supersingular $\ell$ -isogeny graphs as $\ell$ varies, thereby establishing novel connections between graph theory, Iwasawa theory, elliptic curves, and Galois representations of newforms, while proposing a conjecture on the Galois orbits of these newforms.

Original authors: Taiga Adachi, Kosuke Mizuno, Ryosuke Murooka, Sohei Tateno

Published 2026-05-25

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Taiga Adachi, Kosuke Mizuno, Ryosuke Murooka, Sohei Tateno

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: A Map, a Ladder, and a Secret Code

Imagine you are a cartographer trying to understand the hidden structure of a mysterious landscape. In this paper, the authors are mapping a landscape made of elliptic curves (a special type of mathematical shape used in cryptography and number theory).

They are looking at a specific type of map called a Supersingular Isogeny Graph (SIG).

The Map (The Graph): Think of the vertices (dots) on this map as different types of elliptic curves. The lines (edges) connecting them represent "bridges" called isogenies that link one curve to another.
The Ladder (The Tower): The authors build a "tower" of these maps. Imagine taking the original map and creating a giant, multi-layered version of it, where each layer is a slightly more complex copy of the one below. This is called a $\mathbb{Z}_p$ -tower.
The Secret Code (The Invariant): As they climb this tower, they are looking for a specific number called the Iwasawa $\lambda$ -invariant. You can think of this number as a "complexity score" or a "density meter" that tells them how tangled or busy the map gets as you go higher up the tower.

The main question the paper asks is: If we change the rules of the map (by changing the prime number $\ell$ ), how does this complexity score behave? Does it jump around randomly, or does it follow a pattern?

The Main Characters

The Graph ( $X(r, \ell)$ ): This is the "Double Supersingular Isogeny Graph." It's like a city where every intersection (curve) has exactly $\ell + 1$ roads leading out of it. The authors study what happens when they build a tower over this city.
The Newforms (The Oracles): In the world of number theory, there are special functions called "newforms." Think of these as crystal balls or oracles. Each newform holds a secret code (a list of numbers called Fourier coefficients).
- The paper discovers a magical link: The shape of the graph (specifically, its "eigenvalues," which are like the graph's natural frequencies) is directly determined by the numbers inside these crystal balls.
The Galois Orbits (The Families): These crystal balls come in families. If you take one crystal ball and apply a "Galois conjugation" (a mathematical shuffling of its numbers), you get a different ball that belongs to the same family. The size of this family is called the orbit size.

The Discovery: The Pattern of Complexity

The authors fix two numbers ( $r$ and $p$ ) and let the third number ( $\ell$ ) vary over many different prime numbers. They want to know: What complexity scores ( $\lambda$ ) can we get?

They found a beautiful, predictable pattern:

The Formula: The complexity score isn't random. It is calculated by a simple formula:
$\lambda = 1 + 2 \times (\text{Sum of the sizes of selected families})$

Imagine you have a menu of different "families" of crystal balls (orbits). You can pick any combination of these families you like.
- If you pick no families, the score is 1.
- If you pick a family of size 1, the score is $1 + 2(1) = 3$ .
- If you pick a family of size 5, the score is $1 + 2(5) = 11$ .
- If you pick a family of size 1 and a family of size 3, the score is $1 + 2(1+3) = 9$ .
The Guarantee: The paper proves that for almost any combination of families you choose, there are infinitely many prime numbers ( $\ell$ ) that will produce exactly that complexity score.
- It's like saying: "If you want a tower with a complexity score of 9, I can guarantee you can find an infinite number of maps that will give you exactly that score."
The Density: They don't just say these numbers exist; they say they are common. In mathematical terms, they have "positive lower density." This means if you looked at all the prime numbers up to a huge number (like a billion), a significant chunk of them would produce the specific complexity score you are looking for. They aren't rare outliers; they are a regular feature of the landscape.

How They Did It (The Magic Trick)

To prove this, the authors used a powerful tool called the Chebotarev Density Theorem.

The Analogy: Imagine the Galois groups (the mathematical structures governing the crystal balls) as a giant machine with many gears. The authors needed to prove that they could turn the gears in a specific way to get the numbers they wanted.
They used a "Big Image Theorem" (a result by other mathematicians) which essentially guarantees that the machine is flexible enough to produce any valid combination of outcomes.
Because the machine is so flexible, they could prove that for any desired combination of "families" (orbits), there is a specific setting (a specific prime $\ell$ ) that makes the graph's complexity score match that combination.

The Conclusion and The Conjecture

The paper concludes with a list of examples (Table 2) showing how different prime numbers ( $r$ ) create different sets of families, which in turn allow for different odd numbers as complexity scores.

They end with a Conjecture (a guess based on strong evidence):

The Guess: It seems likely that every single odd number (1, 3, 5, 7, 9, etc.) can be realized as a complexity score for some map.
The Implication: If this guess is true, it means the world of these mathematical maps is incredibly rich and diverse. No matter what odd "complexity score" you dream up, there is a mathematical universe out there where that score is the rule.

Summary in One Sentence

The authors proved that the "complexity" of certain mathematical towers built over elliptic curve maps is not random, but is strictly determined by the sizes of families of special number-theoretic functions, and that every possible combination of these families appears frequently as you vary the map's parameters.

Technical Summary: Distributions of Iwasawa $\lambda$ -invariants of $\mathbf{Z}_p$ -towers over supersingular isogeny graphs

Problem Statement

This paper investigates the distribution of Iwasawa $\lambda$ -invariants associated with constant $\mathbf{Z}_p$ -towers over double supersingular $\ell$ -isogeny graphs (DSIGs). The study sits at the intersection of graph theory, Iwasawa theory, and the arithmetic of elliptic curves.

The central object of study is the supersingular $\ell$ -isogeny graph, denoted $SI(r, \ell)$ , defined over the algebraic closure of a finite field $\mathbf{F}_r$ (where $r$ and $\ell$ are distinct primes). The vertices represent isomorphism classes of supersingular elliptic curves, and edges represent $\ell$ -isogenies. The authors consider the "double" version, $X(r, \ell)$ , which is an undirected graph constructed by treating the directed edges of $SI(r, \ell)$ and their inverses as distinct edges.

The problem focuses on the behavior of the Iwasawa $\lambda$ -invariant, $\lambda_\ell$ , associated with a specific $\mathbf{Z}_p$ -tower (a compatible system of $\mathbf{Z}/p^n\mathbf{Z}$ -covers) over $X(r, \ell)$ . Specifically, the authors fix the prime $r$ (the level of the modular curve) and the prime $p$ (the characteristic of the tower), and let the prime $\ell$ vary over infinitely many primes. The goal is to determine the statistical distribution of the values taken by $\lambda_\ell$ as $\ell$ varies.

Methodology

1. Graph-Theoretic Iwasawa Theory

The authors utilize the graph-theoretic analogue of Iwasawa theory developed by Gonet and Vallières. For a $\mathbf{Z}_p$ -tower over a graph $X$ , the complexity $\kappa(X_n)$ (the number of spanning trees of the $n$ -th layer) satisfies an asymptotic formula:
$\text{ord}_p(\kappa(X_n)) = \mu p^n + \lambda n + \nu$
for sufficiently large $n$ . The integers $\mu$ and $\lambda$ are the Iwasawa invariants. The $\lambda$ -invariant is determined by the characteristic element $\Delta(T)$ of the tower, which is derived from the adjacency matrix of the base graph and the voltage assignment defining the tower.

2. Spectral Properties of Isogeny Graphs

A crucial link is established between the adjacency matrix of the supersingular isogeny graph and modular forms. Following Eichler, the eigenvalues of the adjacency matrix $A(SI(r, \ell))$ correspond to the $\ell$ -th Fourier coefficients of weight 2 modular forms of level $r$ . Specifically, the spectrum consists of:

The eigenvalue $\ell+1$ , corresponding to the Eisenstein series $\mathcal{E}_2$ .
The eigenvalues $a_\ell(f)$ , where $f$ ranges over the newforms in the space $S_2(\Gamma_0(r))$ .

The authors define a "constant $\mathbf{Z}_p$ -tower" over $X(r, \ell)$ using a specific voltage assignment (mapping forward edges to $1$ and backward edges to $-1$ in $\mathbf{Z}_p$ ). They derive a formula for the characteristic element $\Delta_\ell(T)$ of this tower, showing it factors into terms related to the eigenvalues of the adjacency matrix.

3. Galois Representations and Big Image Theorems

To analyze the distribution of $\lambda_\ell$ as $\ell$ varies, the authors must control the $p$ -adic valuations of the terms $\ell+1 - a_\ell(f)$ . This requires understanding the distribution of the Fourier coefficients $a_\ell(f)$ modulo $p$ .

The paper employs the "Big Image Theorem" for Galois representations attached to newforms, as established by Lei, Loeffler, and Zerbes. For a set of newforms $f_1, \dots, f_s$ (representatives of Galois orbits) with no complex multiplication, the image of the product of their associated Galois representations (along with the cyclotomic character) is "large" (essentially the full product of general linear groups over the relevant rings) for all but finitely many primes $p$ .

4. Chebotarev Density Theorem

By combining the spectral decomposition of the characteristic element with the Big Image Theorem, the authors construct a specific subset of the Galois group of a finite extension $K/\mathbf{Q}$ . They apply the Chebotarev Density Theorem to show that the Frobenius elements $\text{Frob}_\ell$ fall into specific conjugacy classes with positive density. These classes correspond to specific congruence conditions on the coefficients $a_\ell(f)$ , which in turn dictate the value of the $\lambda$ -invariant.

Key Contributions and Results

1. Explicit Formula for $\lambda_\ell$

The authors derive an explicit formula for the Iwasawa $\lambda$ -invariant of the constant $\mathbf{Z}_p$ -tower over $X(r, \ell)$ . Assuming $r \equiv 1 \pmod{12}$ and $\ell \equiv 1 \pmod{p}$ , they prove:
$\lambda_\ell = 1 + 2 \sum_{k \in W} |O(k)|$
where $W$ is a subset of the indices of the newform orbits, and $|O(k)|$ is the size of the $k$ -th orbit (equal to the degree of the Hecke field of the newform). The term $1$ arises from the Eisenstein series component, and the factor of $2$ arises from the structure of the double graph.

2. The Distribution Theorem (Main Theorem 1.2 / 5.5)

The primary result states that for a fixed prime $r \equiv 1 \pmod{12}$ and for all but finitely many odd primes $p$ , the set of primes $\ell$ for which $\lambda_\ell$ takes a specific value has positive lower density.

Specifically, for any subset $W \subset \{1, \dots, s\}$ of the newform orbits, the set:
$\mathcal{L}_W = \left\{ \ell : \ell \neq r, \ell \equiv 1 \pmod{p}, \lambda_\ell = 1 + 2 \sum_{k \in W} |O(k)| \right\}$
has a positive lower density. This implies that every possible value of the form $1 + 2 \sum |O(k)|$ is realized by infinitely many primes $\ell$ .

3. Connection to Newform Orbits

The paper demonstrates that the possible values of the Iwasawa $\lambda$ -invariant are strictly determined by the arithmetic of the newforms of level $r$ . The size of the newform orbits (degrees of Hecke fields) directly dictates the possible values of $\lambda_\ell$ .

4. Conjecture on Realizability of Odd Integers

Based on computational data from the LMFDB (L-functions and Modular Forms Database), the authors observe that for various $r$ , the set of values $\{1 + 2 \sum_{k \in W} |O(k)|\}$ covers many odd integers. They propose Conjecture 5.6: For any non-negative integer $n$ , there exists a prime $r$ and a subset of newform orbits such that $n = \sum |O(k)|$ . If true, this would imply that every odd integer can be realized as an Iwasawa $\lambda$ -invariant for some constant $\mathbf{Z}_p$ -tower over a DSIG.

Significance and Claims

The paper claims to reveal deep connections between four distinct areas:

Graph Theory: Specifically, the spectral theory of isogeny graphs.
Iwasawa Theory: The study of invariants in towers of graphs.
Elliptic Curves: The arithmetic of supersingular curves and their isogenies.
Galois Representations: The distribution of Fourier coefficients of newforms.

The authors emphasize that their work provides a concrete mechanism (via the Chebotarev density theorem applied to Galois representations of newforms) to predict the distribution of Iwasawa invariants in a graph-theoretic setting. They note that while previous works (e.g., Lei and Müller) studied $\mathbf{Z}_p$ -towers of isogeny graphs, this paper specifically addresses the "constant" towers and establishes a precise distribution theorem linking the invariants to the Galois orbits of newforms.

The paper is modest in its scope, restricting the main theorem to the case where $r \equiv 1 \pmod{12}$ to ensure the symmetry of the adjacency matrix and the absence of certain supersingular $j$ -invariants (0 and 1728). The authors suggest that extending the theory to remove this assumption is a natural direction for future work, which would potentially allow the realization of every odd number as a $\lambda$ -invariant.

The paper also acknowledges related work by Munier and Shnidman regarding the distribution of $p$ -Sylow subgroups of Jacobians of isogeny graphs, noting that clarifying the precise relationship between their results and the Iwasawa-theoretic viewpoint presented here is an interesting open problem.