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Iwasawa theory for abelian towers of digraphs

This paper establishes Iwasawa main conjectures for the Picard and Bowen–Franks groups in $\mathbb{Z}_p^d$ -towers of digraphs by relating their $\ell$ -parts to $p$ -adic $L$ -functions, generalizing classical growth results on ideal class groups, and introducing the concept of defect to analyze the asymptotic behavior of algebraic and analytic ranks.

Original authors: Antonio Lei, Katharina Müller

Published 2026-01-28

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CC BY 4.0

Original authors: Antonio Lei, Katharina Müller

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

Imagine a world made of directed graphs (or "digraphs"). Think of these not as static pictures, but as intricate mazes where every path has a specific direction, like a one-way street system in a giant city. In this paper, the authors, Antonio Lei and Katharina Müller, explore what happens when you build an infinite, layered tower of these mazes, where each new layer is a more complex, detailed version of the one before it.

Here is a breakdown of their journey, translated into everyday language.

1. The Setup: Building a Tower of Mazes

Imagine you have a simple map of a city (a digraph). Now, imagine you want to create a "super-map" that covers every possible route in the original city, but with extra details. You do this repeatedly:

Layer 0: The original city.
Layer 1: A map that covers Layer 0, but with some paths split into two or three.
Layer 2: An even more detailed map covering Layer 1.
...and so on, forever.

The authors call this a $\mathbb{Z}_p^d$ -tower. It's like a fractal of mazes, growing infinitely upward. The "voltage assignment" mentioned in the paper is essentially a rulebook that tells you how to build each new layer from the old one. It's like a set of instructions: "If you take a left turn on the bottom layer, you must take a left-left turn on the next layer."

2. The Two Main Characters: Picard Groups and Bowen–Franks Groups

In this mathematical world, every maze has two special "scorecards" that tell us about its structure.

The Picard Group (The "Tree Count"):
Think of this as counting how many ways you can build a "skeleton" of the city using only the roads, such that you can reach every building without ever getting stuck in a loop. In math terms, this is the number of spanning trees. The authors track how this number changes as the tower grows.
- The Analogy: Imagine you are a city planner trying to lay down power lines to every house. The Picard group counts all the different valid ways you could do this without creating a short circuit (a loop).
The Bowen–Franks Group (The "Shift Space"):
This group is related to how the maze behaves when you look at it as a sequence of moves (like a video game level). It helps mathematicians determine if two different mazes are essentially the same "shape" even if they look different on paper.
- The Analogy: Imagine two different video games. One has a dragon at the start, the other has a robot. But if the rules for moving through the levels are identical, they are "conjugate." The Bowen–Franks group is the tool that checks if the underlying rules are the same.

3. The Main Discovery: The "Main Conjecture"

The core of the paper is proving a Main Conjecture. In simple terms, this is a bridge connecting two different ways of looking at the same problem:

The Algebraic Side: Counting the actual structures (the trees and the shift spaces) in the towers.
The Analytic Side: Using a special "p-adic L-function." Think of this as a magic formula or a "weather forecast" for the graph. It's a function that predicts the behavior of the graph based on the rules used to build the tower.

The Big Claim: The authors prove that the "weather forecast" (the L-function) perfectly predicts the "actual weather" (the size of the Picard and Bowen–Franks groups). If you know the formula, you know exactly how the groups will grow. This is a huge deal because it links a counting problem (algebra) with a formula problem (analysis).

4. Predicting Growth: The "Sinnott–Washington" Theorem

The authors also look at how big these groups get as the tower gets taller.

The Old Way: For simple towers (1D), mathematicians already knew the groups grew in a predictable pattern (like a straight line or a curve).
The New Way: The authors show that for complex, multi-dimensional towers, the growth follows a specific, predictable formula involving powers of $p$ .
The Metaphor: Imagine you are stacking bricks. You might expect the stack to grow linearly. The authors prove that even in a complex, twisting tower, the number of bricks follows a very specific mathematical rhythm, and they can calculate exactly how many bricks you'll have at any height.

5. The "Defect": When the Math Doesn't Match the Reality

Finally, the authors introduce a concept called the Defect.

The Idea: Sometimes, the "analytic" prediction (the formula) suggests a certain amount of complexity, but the "algebraic" reality (the actual count) is slightly different.
The Analogy: Imagine a recipe (the formula) that says a cake should rise 10 inches. But when you bake it, it only rises 8 inches. The "defect" is the difference (2 inches).
The Finding: The authors study how this "gap" behaves as the tower grows. They find that in many cases, this gap either stays the same or grows in a very controlled way.
- They even look at a special case involving elliptic curves (a type of number theory object related to cryptography). In this specific case, they prove the "defect" never changes—it stays constant from the bottom of the tower to the top.

Summary

In short, this paper builds a mathematical bridge between counting structures in infinite, layered directed graphs and predicting formulas that describe them.

They proved that the "magic formulas" (L-functions) perfectly describe the "actual counts" (Picard and Bowen–Franks groups).
They showed how these counts grow as the tower gets taller.
They introduced a "defect" metric to measure the difference between the formula's prediction and reality, proving that in many cases, this difference is stable and predictable.

It's a story about finding order and rhythm in an infinitely complex, growing maze.

Technical Summary: Iwasawa Theory for Abelian Towers of Digraphs

Problem Statement
This paper formulates and proves Iwasawa main conjectures for the Picard groups and Bowen–Franks groups within the context of $\mathbb{Z}_p^d$ -towers of directed graphs (digraphs). While classical Iwasawa theory studies the growth of ideal class groups in towers of number fields, and recent work has extended this to undirected graphs, this work addresses the more general setting of digraphs. The authors aim to connect the algebraic structure of these groups (specifically their $\ell$ -primary parts) to analytic objects defined via voltage assignments, specifically $p$ -adic $L$ -functions. A secondary goal is to generalize classical results by Sinnott and Washington regarding the growth of class groups to this graph-theoretic setting and to investigate the asymptotic behavior of a "defect" measuring the discrepancy between algebraic and analytic ranks.

Methodology
The authors employ a synthesis of algebraic graph theory, Iwasawa theory, and symbolic dynamics.

Algebraic Framework: The study centers on a tower of digraphs $X = X_0 \leftarrow X_1 \leftarrow \dots$ where the Galois group of the cover $X_n \to X$ is isomorphic to $(\mathbb{Z}/p^n\mathbb{Z})^d$ . These towers are realized via $\mathbb{Z}_p^d$ -valued voltage assignments $\alpha$ on the base graph $X$ .
Algebraic Objects: The authors define the inverse limits of the $\ell$ -primary parts of the Picard groups ( $Pic_\ell(X_\infty)$ ) and Bowen–Franks groups ( $BF_\ell(X_\infty)$ ). These are treated as modules over the Iwasawa algebra $\Lambda_\ell(\Gamma) = \mathbb{Z}_\ell[[\mathbb{Z}_p^d]]$ .
Analytic Objects: Two $p$ -adic functions are introduced: the $p$ -adic zeta function $L_p(X, \alpha)$ and the $p$ -adic Bowen–Franks function $\mathcal{L}_p(X, \alpha)$ . These are defined as determinants involving the degree matrix and a twisted adjacency matrix $A_\alpha$ within the ring of polynomials with exponents in $\mathbb{Z}_p$ .
Characteristic Ideals:
- For the case $\ell = p$ , the characteristic ideals are defined via the structure theorem for finitely generated torsion modules over $\Lambda_p(\Gamma)$ .
- For the case $\ell \neq p$ , the authors utilize the framework developed by Bandini and Longhi [BL23] to define characteristic ideals for modules over $\Lambda_\ell(\Gamma)$ , which decomposes based on finite-order characters of $\Gamma$ .
Growth Analysis: To analyze the growth of the orders of these groups, the authors generalize Sinnott's method [Sin87] to the multi-variable setting. This involves estimating the $\ell$ -adic valuation of polynomials evaluated at $p$ -power roots of unity.

Key Contributions and Results

Iwasawa Main Conjectures (Theorem A): The primary result establishes an equality between the characteristic ideal of the algebraic modules and the ideal generated by the analytic $p$ -adic functions. Specifically, for all primes $\ell$ :
$\text{Char}_{\Lambda_\ell(\Gamma)}(Pic_\ell(X_\infty)) = (L_p(X, \alpha))$
$\text{Char}_{\Lambda_\ell(\Gamma)}(BF_\ell(X_\infty)) = (\mathcal{L}_p(X, \alpha))$
The authors prove that $Pic_\ell(X_\infty)$ is a torsion module, while $BF_\ell(X_\infty)$ is torsion if and only if the analytic function is non-zero. This generalizes previous results for undirected graphs and the case $\ell=p$ .
Generalized Growth Theorem (Theorem B): The paper proves a Sinnott–Washington type theorem for $\ell \neq p$ . It establishes that the order of the torsion part of the Picard and Bowen–Franks groups grows asymptotically as:
$\text{ord}_\ell(|Pic_\ell(X_n)_{tor}|) = p^{nd} \mu_\ell(L_p(X, \alpha)) + O(p^{(d-1)n})$
where $\mu_\ell$ denotes the $\ell$ -adic valuation of the leading coefficient of the $p$ -adic function. This generalizes classical results on ideal class groups to digraphs for arbitrary $d \geq 1$ .
Defect Analysis (Theorem C): The authors define the "defect" $\delta(X) = a(X) - b(X)$ , where $a(X)$ is the analytic Bowen–Franks rank (order of the pole of the zeta function at 1) and $b(X)$ is the algebraic rank (rank of the Bowen–Franks group). They prove that in a $\mathbb{Z}_p^d$ -tower, the defect is non-decreasing: $\delta(X_n) \geq \delta(X_{n-1})$ .
- In the case $d=1$ , if $L_p(X, \alpha) \neq 0$ , the defect is shown to be bounded and eventually constant.
- For towers arising from constant voltage assignments, the defect becomes constant for sufficiently large $n$ .
- For isogeny graphs of supersingular elliptic curves, the defect is constant for all $n$ .

Significance and Claims
The paper claims to provide the first formulation of Iwasawa main conjectures for the $\ell$ -primary parts of Picard and Bowen–Franks groups in the setting of directed graphs, bridging the gap between the theory of Ihara zeta functions and the module structure of these groups.

The authors highlight that their generalization of Sinnott's method to multi-variable polynomials (Corollary 6.6) may be of independent interest for other applications involving $\ell$ -adic valuations in multi-variable Iwasawa theory. Furthermore, the introduction of the "defect" concept draws a parallel to the Leopoldt defect in number theory. While the authors note that the conceptual relation between the graph-theoretic defect and the Leopoldt defect is currently unknown, they observe that the behavior of the defect in these towers (increasing and uniformly bounded) mirrors the behavior of the Leopoldt defect in number field towers.

The work relies on recent developments in the theory of modules over Iwasawa algebras (specifically Bandini–Longhi) and connects these algebraic structures to symbolic dynamics, where Bowen–Franks groups serve as invariants for shift spaces. The paper does not propose new experimental applications but rather establishes a rigorous theoretical framework for understanding the arithmetic of graph towers.