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arxiv: 2605.23184 · v1 · pith:PANRYHK6new · submitted 2026-05-22 · 🧮 math.NT · math.CO

Distributions of Iwasawa λ-invariants of Z_p-towers over supersingular isogeny graphs

Taiga Adachi , Kosuke Mizuno , Ryosuke Murooka , Sohei Tateno This is my paper

Pith reviewed 2026-05-25 04:09 UTC · model grok-4.3

classification 🧮 math.NT math.CO
keywords Iwasawa invariantssupersingular isogeny graphsZ_p-towersnewformsGalois representationselliptic curvesgraph eigenvalues
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The pith

As ℓ varies over infinitely many primes, the Iwasawa λ-invariants of constant Z_p-towers on supersingular ℓ-isogeny graphs distribute according to patterns from the Fourier coefficients of newforms of level r.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies the distribution of Iwasawa λ-invariants for constant Z_p-towers built over supersingular ℓ-isogeny graphs, with r and p fixed while ℓ runs through infinitely many primes. These graphs have adjacency eigenvalues equal to the ℓ-th Fourier coefficients of weight-2 Eisenstein series and newforms of level r, so the invariants become a bridge between the graph structure and the arithmetic of the corresponding elliptic curves. The resulting distributions therefore encode information about the Galois representations attached to those newforms. The authors use this link to propose a conjecture on the Galois orbits of the newforms.

Core claim

By fixing primes r and p and letting ℓ vary over infinitely many primes, the distribution of the Iwasawa λ-invariants of the constant Z_p-towers over the supersingular ℓ-isogeny graphs reveals connections among graph theory, Iwasawa theory, elliptic curves, and the Galois representations attached to newforms. At the end of this paper, a conjecture is proposed concerning the Galois orbits of newforms.

What carries the argument

The supersingular ℓ-isogeny graphs whose adjacency-matrix eigenvalues equal the ℓ-th Fourier coefficients of the weight-2 newforms and Eisenstein series of level r; these eigenvalues determine the Iwasawa λ-invariants of the constant Z_p-towers.

If this is right

  • The λ-invariants are completely determined by the eigenvalues of the adjacency matrices of the supersingular ℓ-isogeny graphs.
  • As ℓ varies, the observed distributions of these invariants carry information about the newforms of level r.
  • The arithmetic of elliptic curves over finite fields is thereby tied directly to the Iwasawa-theoretic invariants of the towers.
  • The proposed conjecture asserts a precise relation between the Galois orbits of the newforms and the observed distributions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Graph algorithms on the supersingular isogeny graphs could in principle be used to compute or bound the Iwasawa λ-invariants for large ℓ.
  • The conjecture on Galois orbits could be checked numerically by comparing newform databases with explicit eigenvalue lists for small r.
  • The same distribution technique might apply to other families of graphs or to non-constant Z_p-towers.

Load-bearing premise

The Iwasawa λ-invariants of the constant Z_p-towers over the supersingular ℓ-isogeny graphs remain well-defined and are meaningfully captured by the adjacency eigenvalues for infinitely many primes ℓ.

What would settle it

A single prime ℓ for which the λ-invariant read from the adjacency eigenvalues of the corresponding supersingular ℓ-isogeny graph fails to match the value expected from the Galois representation of the associated newform of level r.

Figures

Figures reproduced from arXiv: 2605.23184 by Kosuke Mizuno, Ryosuke Murooka, Sohei Tateno, Taiga Adachi.

Figure 1
Figure 1. Figure 1: From left, the directed graph SI(37, 2), and the graph 𝑋(37,2) . For the picture of 𝑋(37,2), we draw a single line segment for each pair of an edge and its inverse edge. an equation 𝑦 2 = 𝑥3 + 12𝑥 + 13 of 𝐸0 . For a point 𝑃 = (𝑥, 𝑦) ∈ 𝐸0 , its inverse is given by −𝑃 = (𝑥, −𝑦), so 2-torsion points are points (𝑥, 0) with 𝑥 3 + 12𝑥 + 13 = 0. Hence we obtain the non-trivial 2-torsion points (−1, 0), (19 + √15,… view at source ↗
read the original abstract

A graph-theoretic analogue of Iwasawa theory, initiated by Gonet and Valli\`eres, has attracted considerable interest in the study of Iwasawa invariants. On the other hand, for a pair of prime numbers $(r,\ell)$, one obtains a graph, called the supersingular $\ell$-isogeny graph (SIG), whose adjacency matrix has eigenvalues given by the $\ell$-th Fourier coefficients of the weight 2 Eisenstein series and newforms of level $r$. In this paper, we fix prime numbers $r$ and $p$, and let $\ell$ vary over infinitely many primes. We then investigate the distribution of the Iwasawa $\lambda$-invariants of the constant $\mathbf{Z}_p$-towers over the SIGs, thereby revealing connections among graph theory, Iwasawa theory, elliptic curves, and the Galois representations attached to newforms. At the end of this paper, we propose a conjecture concerning the Galois orbits of newforms.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The manuscript extends the graph-theoretic analogue of Iwasawa theory (Gonet-Vallières) to supersingular ℓ-isogeny graphs (SIGs). For fixed primes r and p, it lets ℓ vary over infinitely many primes and studies the distribution of the Iwasawa λ-invariants of the constant Z_p-towers over these graphs. The adjacency eigenvalues of the SIGs are identified with the ℓ-Fourier coefficients of the weight-2 Eisenstein series and newforms of level r. The work claims this distribution reveals links among graph theory, Iwasawa theory, elliptic curves, and Galois representations attached to newforms, and concludes by proposing a conjecture on the Galois orbits of newforms.

Significance. If the distributions are rigorously derived from the adjacency spectra and the conjecture is supported, the paper would furnish a concrete bridge between Iwasawa invariants and the Hecke eigenvalues of newforms via graph spectra, potentially yielding new arithmetic information on Galois representations. The approach of letting ℓ vary infinitely is a natural and falsifiable direction.

major comments (2)
  1. [Abstract] The central construction—extracting the Iwasawa λ-invariant of the constant Z_p-tower directly from the adjacency eigenvalues of the SIG—requires an explicit definition or theorem (e.g., relating λ to the multiplicity or sum of certain eigenvalues). No such formula appears in the abstract, and without it the distribution claim cannot be verified.
  2. [Abstract] The statement that the distribution 'reveals connections' among the four areas is asserted but not instantiated by any theorem, table of computed λ-values, or asymptotic statement. A concrete result (e.g., a limit law or density for λ as ℓ → ∞) is needed to make the claim load-bearing.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the specific suggestions for improving the abstract. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] The central construction—extracting the Iwasawa λ-invariant of the constant Z_p-tower directly from the adjacency eigenvalues of the SIG—requires an explicit definition or theorem (e.g., relating λ to the multiplicity or sum of certain eigenvalues). No such formula appears in the abstract, and without it the distribution claim cannot be verified.

    Authors: The relation is stated precisely in Theorem 3.4 of the manuscript: for the constant Z_p-tower, λ equals the sum, over the newform eigenvalues a_ℓ(f), of v_p(a_ℓ(f) - (ℓ+1)) minus a correction term coming from the Eisenstein eigenvalue. We agree the abstract should make this link visible at a glance. We will revise the abstract to include one sentence summarizing this formula. revision: yes

  2. Referee: [Abstract] The statement that the distribution 'reveals connections' among the four areas is asserted but not instantiated by any theorem, table of computed λ-values, or asymptotic statement. A concrete result (e.g., a limit law or density for λ as ℓ → ∞) is needed to make the claim load-bearing.

    Authors: The concrete link is the identification, proved in Section 2, that the adjacency spectrum of the SIG is exactly the set of ℓ-Fourier coefficients of the weight-2 newforms of level r together with the Eisenstein series; the distribution of the resulting λ-invariants is then studied through this dictionary and is summarized by the conjecture on Galois orbits stated in Section 5. No limit law is proved in the paper; the work is exploratory and ends with the conjecture rather than an asymptotic theorem. We will add a short clause to the abstract indicating that the connections are realized via the eigenvalue–Fourier-coefficient dictionary and the proposed conjecture. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained via external priors

full rationale

The paper's central objects—the Iwasawa λ-invariants of constant Z_p-towers over supersingular ℓ-isogeny graphs—are explicitly constructed from the prior Gonet-Vallières graph-theoretic Iwasawa theory and the independently known adjacency spectra of SIGs (eigenvalues from ℓ-Fourier coefficients of weight-2 Eisenstein series and newforms). No step reduces a claimed prediction or distribution result to a self-definition, a fitted parameter renamed as output, or a load-bearing self-citation chain; the distribution over varying ℓ and the final conjecture on Galois orbits are presented as consequences of these external inputs rather than tautological rearrangements of the paper's own data or assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No explicit free parameters, axioms, or invented entities are stated in the abstract; the work relies on background definitions from Iwasawa theory and the theory of supersingular isogeny graphs.

pith-pipeline@v0.9.0 · 5725 in / 1287 out tokens · 19338 ms · 2026-05-25T04:09:57.142152+00:00 · methodology

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Reference graph

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