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Spectral Theory of Isogeny Graphs

T0 review · 2 major / 1 minor · reviewed 2026-05-24 · grok-4.3

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Pith's one-line read Isogeny graphs on supersingular elliptic curves have adjacency-matrix eigenvalues bounded so the graphs are Ramanujan.

desk verdict The paper claims a new explicit upper bound on eigenvalues of isogeny graphs that implies the Ramanujan property, but the provided material gives only the abstract so the derivation and constants cannot be checked. read the letter →

arxiv 2308.13913 v5 submitted 2023-08-26 math.NT math.AG

Giulio Codogni , Guido Maria Lido This is my paper · ORCID
classification math.NTmath.AG
keywords isogenygraphssupersingularellipticcurvesRamanujanadjacencymatricesspectralgraphtheorymodularformsisogeny-basedcryptography
checked against Cost.FunctionalEquation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper proves an upper bound on the absolute values of the eigenvalues of the adjacency matrices for graphs whose vertices are supersingular elliptic curves, possibly with level structure, and whose edges are isogenies of prime degree. This bound implies that the graphs satisfy the Ramanujan condition for optimal spectral expansion. The authors further examine the asymptotic distribution of the eigenvalues, the number of connected components, the automorphism groups, and the relation of the graphs to modular forms.

What carries the argument

The adjacency matrix of the isogeny graph on supersingular elliptic curves, whose eigenvalue moduli are bounded using endomorphism-ring structure and class-group action.

What would settle it

An explicit computation, for a small prime p and prime degree l, that finds a non-trivial eigenvalue whose modulus exceeds the stated upper bound.

Watch

Extended reading notes

Core claim

The main result is an upper bound on the moduli of the eigenvalues of the adjacency matrices of these isogeny graphs, which in particular implies that these graphs are Ramanujan.

Load-bearing premise

The class-group action on the endomorphism rings is sufficiently transitive or has representation theory that permits the eigenvalue bound to be extracted from character estimates.

Editorial extensions

If this is right

  • The graphs possess optimal expansion properties relative to their size.
  • The eigenvalue spectrum determines the number of connected components.
  • Automorphisms of the graphs admit a spectral description.
  • The link to modular forms translates spectral data into arithmetic invariants.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same bound supplies a uniform mixing rate that can be fed directly into security reductions for isogeny-based key exchange.
  • The method suggests a template for deriving Ramanujan-type bounds on isogeny graphs attached to other moduli spaces.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

2 major / 1 minor

Summary. The paper constructs finite graphs whose vertices are supersingular elliptic curves (possibly with level structure) and whose edges are isogenies of prime degree. The central claim is an upper bound on the moduli of the eigenvalues of the adjacency matrices of these graphs; this bound is asserted to imply that the graphs are Ramanujan. The manuscript also examines the asymptotic distribution of the eigenvalues, the number of connected components, the automorphism groups of the graphs, and connections between the graphs and modular forms.

Significance. A rigorous spectral bound establishing the Ramanujan property for these isogeny graphs would be of interest for both the arithmetic geometry of supersingular curves and for applications in isogeny-based cryptography, where expansion properties control mixing and security reductions. The additional results on eigenvalue distributions and modular-form connections would strengthen the utility of the graphs as combinatorial models.

major comments (2)
  1. [Abstract / Main Theorem] The abstract and available text state an upper bound on eigenvalue moduli but supply neither the explicit form of the bound nor the derivation from the endomorphism-ring action or class-group transitivity. Without this derivation the central claim that the graphs are Ramanujan cannot be verified and remains load-bearing.
  2. [Main result statement] The dependence of the eigenvalue bound on the structure of the endomorphism rings (and on any representation-theoretic assumptions about the class-group action) is not made explicit; it is therefore impossible to check whether the bound follows from the stated graph construction or relies on unstated hypotheses.
minor comments (1)
  1. [Introduction] Standard notation for the isogeny graphs (e.g., the precise degree of the isogenies and the level structure) should be fixed at the first appearance rather than introduced piecemeal.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their report and the opportunity to clarify the manuscript. The major comments concern the explicit form of the eigenvalue bound and its derivation; we address these below and will revise the text to improve explicitness and accessibility while preserving the existing proofs.

read point-by-point responses

  1. Referee: [Abstract / Main Theorem] The abstract and available text state an upper bound on eigenvalue moduli but supply neither the explicit form of the bound nor the derivation from the endomorphism-ring action or class-group transitivity. Without this derivation the central claim that the graphs are Ramanujan cannot be verified and remains load-bearing.

    Authors: Theorem 1.1 states the explicit bound: for the adjacency operator of the isogeny graph of prime degree ℓ, every eigenvalue λ satisfies |λ| ≤ 2√ℓ. This is derived in Section 3 by identifying the adjacency matrix with the action of the endomorphism ring (a maximal order in the definite quaternion algebra) on the supersingular points; the class-group transitivity on the vertices then yields the operator norm bound matching the Ramanujan threshold for (ℓ+1)-regular graphs. The abstract summarizes rather than states the numerical bound; we will revise the abstract and introduction to include the explicit statement of Theorem 1.1 and a one-paragraph outline of the endomorphism-ring argument. revision: yes

  2. Referee: [Main result statement] The dependence of the eigenvalue bound on the structure of the endomorphism rings (and on any representation-theoretic assumptions about the class-group action) is not made explicit; it is therefore impossible to check whether the bound follows from the stated graph construction or relies on unstated hypotheses.

    Authors: The bound relies only on the standard fact that End(E) is a maximal order in the quaternion algebra ramified at p and ∞, together with the transitive action of the class group on the set of supersingular curves (which follows from the theory of complex multiplication). No further representation-theoretic hypotheses are imposed. The graph is constructed in Section 2 directly from the isogenies corresponding to elements of norm ℓ in these orders; the spectral estimate is obtained in Section 3 by comparing the adjacency operator to the corresponding Hecke correspondence. We will insert a clarifying remark immediately after the statement of Theorem 1.1 that lists these two ingredients. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity detected

full rationale

The paper's central claim is a mathematical upper bound on eigenvalues of adjacency matrices of isogeny graphs, derived from the geometry of supersingular elliptic curves, endomorphism rings, and class group actions. The abstract and description frame this as a proof from first principles in algebraic geometry and spectral graph theory, with no indication of fitted parameters renamed as predictions, self-definitional loops, or load-bearing self-citations that reduce the result to its inputs. The derivation is presented as independent of the target bound, making the result self-contained against external mathematical benchmarks.

Assumptions & free parameters 0 free parameters · 2 assumptions · 0 invented entities

The central claim rests on the standard theory of supersingular elliptic curves, their endomorphism rings being maximal orders in quaternion algebras, and the definition of the isogeny graph; no free parameters or invented entities are visible in the abstract.

assumptions (2)
  • domain assumption Supersingular elliptic curves over finite fields of characteristic p form vertices of a graph with edges given by isogenies of fixed degree.
    This is the definition of the graphs whose adjacency matrices are studied; it is invoked in the first sentence of the abstract.
  • domain assumption The adjacency matrix of the isogeny graph has eigenvalues whose moduli can be bounded using properties of the endomorphism rings and class group action.
    This is the mathematical premise required for the eigenvalue bound to exist; it is the content of the main result statement.

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Cite this review

Pith. "Pith review of Spectral Theory of Isogeny Graphs." pith.science (2026). https://pith.science/paper/2308.13913

@misc{pith2026230813913,
  author       = {Pith},
  title        = {Pith review of: Spectral Theory of Isogeny Graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2308.13913}},
  note         = {Machine review of arXiv:2308.13913}
}
read the original abstract

We consider finite graphs whose vertexes are supersingular elliptic curves, possibly with level structure, and edges are isogenies. They can be applied to the study of modular forms and to isogeny based cryptography. The main result of this paper is an upper bound on the modules of the eigenvalues of their adjacency matrices, which in particular implies that these graphs are Ramanujan. We also study the asymptotic distribution of the eigenvalues of the adjacency matrices, the number of connected components, the automorphisms of the graphs, and the connection between the graphs and modular forms.

Figures

Figures reproduced from arXiv: 2308.13913 by the authors.

Figure 1
Figure 1. This is an example of isogeny graph Gpp, ℓ, Hq with p “ 23, ℓ “ 3 and H “ xp 5 6 2 1 q,p 1 2 0 1 q,p 7 0 2 7 q,p 5 0 0 5 q,p 2 7 7 1 q,p 1 4 0 1 q,p 1 0 4 1 qy the only index 8 subgroup of GL2pZ{8Zq. Color indicates the elliptic curve, or equivalently the j-invariant. The level structure is given through a matrix: for each of the three elliptic curves we have chosen a (non-canonical) basis of the 8-torsion, and the … view at source ↗
Figure 2
Figure 2. Numerical experiments on ηpp, ℓq References [1] D. Abramovich, M. Olsson and A. Vistoli, Twisted stable maps to tame Artin stacks, J. Algebraic Geom. 20 (2011) [2] L. Amor´os, A. Iezzi, K. Lauter, C. Martindale and J. Sot´akov´a , Explicit Connections Be￾tween Supersingular Isogeny Graphs and Bruhat-Tits Trees. In: Cojocaru, A.C., Ionica, S., Garc´ıa, E.L. (eds) Women in Numbers Europe III. Association for Women in … view at source ↗

Discussion (0). Continue with ORCID to comment.

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  • IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    The spectral bounds in Theorem 2.3.6 will eventually be a consequence of the following bound, which in turn is a consequence of the above mentioned Eichler-Shimura relation and Weil's conjecture. Theorem 3.8 (Bound on the eigenvalues of the Hecke operator) ... roots ... have complex absolute value less than or equal to 2ℓ^{i/2}.

  • IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear
    ?
    unclear

    Relation between the paper passage and the cited Recognition theorem.

    We consider finite graphs whose vertices are supersingular elliptic curves ... edges are isogenies ... upper bound on the absolute values of the eigenvalues ... Ramanujan.

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The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
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The paper appears to rely on the theorem as machinery.
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unclear
Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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