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REVIEW 3 major objections 4 minor 95 references

This paper classifies all isogeny graphs of non-CM elliptic curves over characteristic-0 fields, showing each graph decomposes into explicit p-primary pieces and each piece belongs to a small, fully realized family of weighted trees.

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2026-08-04 06:11 UTC pith:JEL2WLA5

load-bearing objection A serious, mostly careful classification of characteristic-0 isogeny graphs that is worth refereeing, with one load-bearing dependency on an external counting lemma that needs scrutiny. the 3 major comments →

arxiv 2608.02494 v1 pith:JEL2WLA5 submitted 2026-08-03 math.NT math.AGmath.CO

Isogeny graphs of elliptic curves in characteristic zero

Alexander J. Barrios , Enrique Gonz\'alez-Jim\'enez , Ivan Novak This is my paper
classification math.NT math.AGmath.CO MSC 11G0511G0711G1511F8014K0205C2505C51
keywords elliptic curvesisogeny graphsGalois representationsp-adic representationsCartesian product of graphsp-blooming invariantmodular curvespotential complex multiplication
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper asks which isogeny graphs—networks whose vertices are elliptic curves isogenous to a given curve and whose edges record prime-degree isogenies—can occur for an elliptic curve without complex multiplication over a field of characteristic zero. It proves that every such graph is a weak Cartesian product of p-primary graphs, one for each prime p, and then classifies each p-primary graph completely. Each p-primary graph is isomorphic to one of the explicit families H^r_{p^k}, H^r_{p^∞}, or H^r_{p^∞,+}; conversely, every graph in these families is realized by some elliptic curve over some characteristic-0 field, with the single exception of the path graphs H^0_{2^k} for k≥2. Because the classification is exhaustive, it settles the long-open question of which isogeny configurations can appear over Q and over other characteristic-0 fields, and it makes the structure of isogeny graphs readable directly from the associated Galois representation.

Core claim

The central claim is Theorem 2 (together with Theorem 1): for an elliptic curve E over a characteristic-0 field K with End_K E ≅ Z, the full isogeny graph G(E/K) is graph-isomorphic to the weak Cartesian product of its p-primary graphs (G_p(E/K), [E]_K). Each p-primary graph is either a finite graph H^r_{p^k} or an infinite graph H^r_{p^∞} or H^r_{p^∞,+}, where r is determined by the p-blooming invariant I_p(E/K), the minimum p-adic valuation of the difference of the two eigenvalues of the p-adic Galois representation. Conversely, every graph in these families occurs, except H^0_{2^k} for k≥2. The proof works by identifying each p-primary graph with a subgroup of GL_2(Z_p) and then analyzing

What carries the argument

The load-bearing objects are the explicit weighted tree families H^r_{p^k} and the infinite families H^r_{p^∞} and H^r_{p^∞,+}, built by repeatedly p-blossoming a path, line, or ray. The main tool is the p-adic Galois representation ρ_{E,p^∞}: G_K → GL_2(Z_p), together with the p-blooming invariant I_p(E/K), which measures how many layers of p-isogenies are forced to be rational. The proof also uses a decomposition theorem for the lattice of cyclic Galois-invariant submodules of the torsion subgroup, which converts the graph decomposition into a statement about primary components of abelian torsion groups.

Load-bearing premise

The classification's completeness rests on the imported claim that a non-CM curve over a characteristic-0 field admits either p^{min{α,⌊j/2⌋}} or 2p^α rational p^j-isogenies; if that dichotomy missed a case, the family would be incomplete.

What would settle it

Compute, for a non-CM elliptic curve over a characteristic-0 field, the number of rational p^j-isogenies for j = 1 through k; any count outside {p^{min(α,⌊j/2⌋)}, 2p^α} would falsify the dichotomy and hence Theorem 2. Equivalently, produce a 2-primary graph with a vertex admitting exactly two rational 2-isogenies and no third, which would realize the supposedly impossible graph H^0_{2^k} for some k≥2.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

  • If the classification is correct, it recovers and fully explains the classical classification of rational isogeny graphs, now as a special case of a characteristic-0 phenomenon.
  • Over fields admitting a real embedding, the isogeny class degree uniquely determines the isogeny graph, so there is no hidden structural ambiguity.
  • The classification gives an algorithm that outputs the pointed isogeny graph from the adelic Galois image of the curve, making the graph effectively computable in practice.
  • For elliptic curves with potential complex multiplication, the p-primary graphs are classified in terms of conductors of endomorphism rings along maximal paths, extending earlier characterizations of n-isogenies.
  • The number of vertices in a finite isogeny class is given by an explicit formula depending only on k and r for each prime p, giving a direct count of curves in the class.

Where Pith is reading between the lines

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

  • A likely consequence is that the same combinatorial classification could be extended to abelian varieties with no additional endomorphisms, since the cyclic-submodule lattice decomposition is stated in that generality.
  • The p-blooming invariant may serve as a practical isogeny-class invariant for computational work over number fields, since it can be approximated from mod p^k data and stabilizes to the full p-adic value.
  • The classification suggests a natural test for completeness: for a given number field K, once the possible Galois images are known, the possible isogeny graphs are determined by intersecting the groups H^r_{p^k} with the image, so the graph classification is equivalent to a Galois-image classification.
  • The exception H^0_{2^k} for k≥2 reflects a local rigidity statement—any curve with two rational 2-isogenies automatically has three—which may have analogues for other primes in more general settings.

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

3 major / 4 minor

Summary. The paper classifies the isogeny graphs of elliptic curves E over characteristic-0 fields K with End_K(E) ≅ Z. Theorem 1 (Theorem 3.8) decomposes G(E/K) as a weak Cartesian product of pointed p-primary graphs. The main classification (Theorem 2; finite case Theorem 4.25, infinite case Theorem 6.15) asserts that every p-primary graph is isomorphic to one of the explicitly constructed families H^r_{p^k}, H^r_{p∞}, or H^r_{p∞,+}, with the parameter r controlled by the new p-blooming invariant I_p(E/K), and that all members occur except H^0_{2^k} for k ≥ 2. The paper further identifies the associated subgroups of GL_2, studies potential-CM isogeny graphs, gives an algorithm for computing the graph from the adelic Galois image, and connects the classification to modular curves and genus-0 parameterizations.

Significance. If correct, the result is a complete and striking classification in a previously unsettled characteristic-0 setting, recovering the rational classification of Chiloyan–Lozano-Robledo and giving new structural invariants. The paper’s strong points include a clean and self-contained proof of the primary decomposition, a detailed combinatorial description of the candidate graph families, an isogeny-class invariant I_p(E/K), and reproducible computational data referenced by the authors. However, the central exhaustiveness claim rests on a counting dichotomy imported from the proof of a result of Novak that is only partially stated in the paper; this dependency must be resolved before the classification can be accepted as fully established.

major comments (3)
  1. [§4.2, proof of Theorem 4.25] The decisive counting step is the assertion, taken from the proof of [70, Proposition 3.1], that the number of K-rational p^j-isogenies of E_0 is either p^{min{α,⌊j/2⌋}} or 2p^α, where α = min v_p(a-c) over the Borel-conjugate mod p^k image. This is stronger than Proposition 2.2, which records only a finite list of possible counts and says nothing about the structural description in terms of α. The subsequent exclusion of the 2p^α branch uses the internal description of that branch in [70], in particular the 'proper solution modulo p^{2r+1}' criterion. If that dichotomy had any additional branch, or if the excluded branch can occur under the paper’s hypotheses, then the family H^r_{p^k} is not exhaustive. The same issue propagates to Lemma 6.14 and Theorem 6.15 for the infinite case. Please state the full dichotomy as a theorem in this paper with proof, or show directly that the weaker P
  2. [§4.2 and §6.2, converses of Theorems 4.25 and 6.15] The existence direction is handled by the sentence 'It then follows from the Galois correspondence' after defining the subgroups H^r_{p^k}, H^r_{p∞}, and H^r_{p∞,+}. This is not a construction: it must be shown that, for arbitrary k,r,p, there is a field K of characteristic 0 and an elliptic curve E/K whose mod p^k or p-adic image has exactly the required shape, including the non-containment conditions (ii)–(iv) in Theorem 4.25. This is a standard type of assertion, but it is load-bearing for the converse part of the main theorem, and the exceptional case H^0_{2^k} for k ≥ 2 is derived only from the observation that v_2(a-c) ≥ 1 for upper-triangular matrices over Z/2^kZ. A more explicit argument, or a precise reference for the Galois-existence step, is needed.
  3. [§6.1, Definition 6.3 and the convention for H^∞_{p∞,+}] The 'skeleton' convention for H^∞_{p∞,+} is nonstandard and is stated negatively in Remark 6.4: the graph is defined as a union of graphs I_j, but no distinguished skeleton is specified. This matters because Lemmas 6.8 and 6.9 assert uniqueness of trees with prescribed bloom depths relative to a line or ray. For r=∞ the ray I_0 is constructed explicitly, so the definition is workable, but the presentation would be clearer if the distinguished ray for H^∞_{p∞,+} were declared as part of the definition rather than inferred.
minor comments (4)
  1. [Abstract] The abstract says every p-primary graph is a member of the families H^r_{p^k} and H^r_{p∞,+}, omitting the family H^r_{p∞}, which appears in the main theorem (Theorem 2). This mismatch should be corrected.
  2. [Throughout] There are several typographical errors: 'assing weights', 'admited', 'the context of Lemma 6.11' should be 'the content', and some cross-references are abbreviated inconsistently ('loc. cit.' without antecedent). These do not affect the mathematics but should be cleaned up.
  3. [§5, Proposition 5.10] The proof of Proposition 5.10 invokes Theorem 6.15, which is proved later in the paper. The text contains a note saying the statement assumes this fact, but for logical hygiene it would be preferable to state Proposition 5.10 as conditional on the infinite classification, or to reorder the sections so that the infinite classification precedes the applications.
  4. [§7, Table 2] The table is introduced with 'The table below summarizes Kwon’s result' before the columns are described, and the fourth column is labelled only as '#m of n-isogenous curves'. It would help to define the notation f, f_E, and the divisibility conventions explicitly in the caption or in the surrounding text.

Circularity Check

0 steps flagged

No circularity: the p-blooming invariant is defined from the Galois image independently of the graph shapes, and the r-formula is proved rather than assumed; the Novak [70] citation is load-bearing for exhaustiveness but is external support, not a fitted input.

full rationale

I find no circular reduction at the level of definitions or fitted parameters. The p-blooming invariant I_p(E/K) (Definition 5.1) is defined solely from the p-adic Galois image, min v_p(λ1−λ2), with no reference to the graphs H^r_{p^k}, H^r_{p∞}, or H^r_{p∞,+}; Corollary 5.7 then proves r = min{I(E),⌊k/2⌋} from the preceding counting argument, so the graph parameter is not built into the invariant. In the finite classification, Theorem 4.25 constructs a k-spine, produces a subgraph isomorphic to H^r_{p^k}, and uses the imported count — “From the proof of [70, Proposition 3.1], we have that the number of p^j-isogenies admitted by E_0 over K is either p^{min{α,⌊j/2⌋}} or 2p^α” — to exclude extra edges. The infinite classification (Lemma 6.14, Theorem 6.15) uses the same dichotomy. This dependency on Novak [70] is genuine and load-bearing for exhaustiveness: if that dichotomy had another branch, the listed families would be incomplete. However, [70] is an external, parameter-free statement about isogeny counts, not a restatement of the graph classification, and the paper does not reduce any equation to its own conclusion. The converse realization via “it then follows from the Galois correspondence” is asserted rather than fully constructed, but that is an omitted proof detail rather than circularity. The correct place for the [70] concern is correctness risk, not a circularity score.

Axiom & Free-Parameter Ledger

0 free parameters · 7 axioms · 2 invented entities

The central claims rest on the stated non-CM condition, the self-cited isogeny-counting dichotomy from [70], and standard background in elliptic curves, Galois representations, and graph theory. No constants are fitted to data; the graph parameters k and r are determined by the theorems (r = min{I_p, ⌊k/2⌋}), and the p-blooming invariant is an intrinsic, independently computable quantity. The main non-background imported input is Novak's counting dichotomy (Prop. 2.2 / [70, Prop. 3.1]).

axioms (7)
  • domain assumption End_K E ≅ Z (the curve has no K-rational endomorphisms beyond Z)
    All main theorems (1, 2, 4.25, 6.15, 8.1) are stated under this hypothesis; Examples 3.13 and 7.3 show Theorem 1 and Corollary 3.9 fail for CM curves, and Lemma 2.1 / Proposition 3.11 use it essentially.
  • domain assumption Novak's isogeny-counting dichotomy: for a non-CM E over char-0 K, the number of K-rational p^j-isogenies is either p^{min{α,⌊j/2⌋}} or 2p^α, with per-degree counts as in Proposition 2.2
    Imported from [70, Props. 1.1, 1.2, 3.1]; used in the proofs of Theorems 4.25 and 6.15 to conclude G_p(E/K) has no vertices beyond a constructed subgraph; exhaustiveness of the entire classification depends on it.
  • standard math Standard elliptic-curve and Galois-representation facts: Weil pairing, Tate module isogeny invariance, determinant equals cyclotomic character, Faltings finiteness over number fields
    Used throughout Sections 2-7; cited to [27,36,40,74,79,80,84] and [30,67].
  • standard math Unique prime factorization for connected graphs under the (weak) Cartesian product (Sabidussi, Vizing, Imrich, Miller)
    Supplies the graph-theoretic framework for Theorem 1 / Proposition 3.7; cited [41,66,76,89].
  • domain assumption A curve admitting two distinct K-rational 2-isogenies admits exactly three
    Used to exclude H^0_{2^k} for k≥2 (Section 4, discussion before Example 4.4); consistent with the p=2 conditions in Theorem 2(a).
  • domain assumption Kwon's characterization of n-isogenies over Q(j(E)) for potential-CM curves, with Bourdon-Clark extensions
    Drives Section 7: Lemma 7.2 and Theorem 7.5 classify G_p(E/Q(j(E))) via the conductor/discriminant table; cited [56], [13], [20], [21].
  • domain assumption Over the algebraic closure the p-primary graph is the Bruhat-Tits tree T_p (homothety classes of lattices in V_p(E))
    Motivates the infinite families H^r_{p∞} and the identification of H^∞_{p∞}; cited [2,16,24,79,80].
invented entities (2)
  • p-blooming invariant I_p(E/K) independent evidence
    purpose: Isogeny-class invariant that determines the shape index r of G_p(E/K) via r = min{I_p, ⌊k/2⌋}; minimal over real-embedding fields
    Defined intrinsically as min_σ v_p(λ1−λ2) over the p-adic Galois image; it is computable, base-change monotone (Cor. 5.5), isogeny-class invariant (Prop. 5.2), and equals 1 (p=2) / 0 (p odd) over real-embedding fields (Prop. 5.10) — all checkable without the classification.
  • Graph families H^r_{p^k}, H^r_{p∞}, H^r_{p∞,+} independent evidence
    purpose: Explicit combinatorial models claimed to exhaust all p-primary isogeny graphs
    Purely combinatorial objects (p-blossoming of paths, lines, rays); vertex counts and distance profiles are computed directly (Cor. 4.22, Lemmas 4.23, 6.11, 6.12), and realizability is demonstrated by exhibiting Galois-image subgroups; the non-realizable member H^0_{2^k} (k≥2) is explained by the three-2-isogenies fact.

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read the original abstract

For an elliptic curve $E$ defined over a field $K$ of characteristic $0$ with $\operatorname{End}_K \! E \cong \mathbb{Z}$, we classify which isogeny graphs $\mathcal{G}(E/K)$ can occur. We first show that $\mathcal{G}(E/K)$ decomposes as a weak Cartesian product of its $p$-primary isogeny graphs, one for each prime $p$, thereby reducing the problem to classifying $p$-primary isogeny graphs. We then show that each such graph is isomorphic, as an edge-weighted graph, to a member of an explicit family of edge-weighted graphs $\mathcal{H}_{p^k}^r$ and $\mathcal{H}_{p^{\infty,+}}^r$, every member of which occurs as a $p$-primary isogeny graph except for $\mathcal{H}_{2^k}^0$ for $k\ge 2$. The proof relies on a detailed study of the $p$-adic Galois representation attached to $E$, through which we identify each graph with a subgroup of $\operatorname*{GL}\nolimits_{2}(\mathbb{Z}_{p})$. More generally, we identify subgroups of $\operatorname*{GL}\nolimits_{2}(\widehat{\mathbb{Z}})$ for each possible isogeny graph and describe their corresponding modular curves, completing, in the genus $0$ case, the explicit parameterization of isogeny graphs via parameterized isogenous families of elliptic curves. We also introduce the $p$-blooming invariant $\mathfrak{I}_p(E/K)$, an isogeny class invariant determining the value of $r$ in the $p$-primary isogeny graph, and show that elliptic curves over fields with a real embedding attain the smallest possible value. As applications, we characterize the isogeny graphs of elliptic curves with potential complex multiplication; give an algorithm for determining the isogeny graph from the adelic Galois representation; recover the classification of rational isogeny graphs; and, under GRH, classify the isogeny graphs occurring over certain number fields.

Figures

Figures reproduced from arXiv: 2608.02494 by Alexander J. Barrios, Enrique Gonz\'alez-Jim\'enez, Ivan Novak.

Figure 1
Figure 1. Figure 1: The graph of H∞ 2∞. Figure reproduced from [18]. Next, consider the increasing sequence H0 p∞ ⊆ H1 p∞ ⊆ H2 p∞ ⊆ · · · . This leads to H∞ p∞ = [ r≥0 Hr p∞. Necessarily, each skeletal vertex has bloom depth ∞. This graph is precisely the p-Bruhat-Tits tree, in which every vertex is p-bloomed. These graphs occur as the primary isogeny graph of non-CM elliptic curves over an algebraically closed field. See [P… view at source ↗

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