Isogeny graphs of abelian varieties and singular ideals in orders

Caleb Springer; Sarah Arpin; Stefano Marseglia

REVIEW 3 major objections 3 minor 30 references

Isogeny graphs of abelian varieties and singular ideals in orders

T0 review · 3 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that isogeny graphs of any-dimensional abelian varieties with commutative endomorphism ring and a fixed locally Bass order are generalized volcanos, covering non-simple and non-ordinary classes.

desk verdict A bold generalization of Kohel's volcano theorem to higher-dimensional abelian varieties, but the abstract leaves the load-bearing ideal-theoretic dictionary unproved, so the full text needs careful checking. read the letter →

arxiv 2508.03570 v1 pith:DWDWHR42 submitted 2025-08-05 math.NT

Sarah Arpin , Stefano Marseglia , Caleb Springer This is my paper · ORCID

classification math.NT MSC 11G1014K02

keywords isogenygraphsvolcanostructureabelianvarietiescommutativeendomorphismringslocallyBassordersoverorderlatticesétalealgebrasnon-ordinaryclasses

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

This paper claims that the classical volcano theorem for isogeny graphs of ordinary elliptic curves extends to abelian varieties of any dimension whose endomorphism ring is commutative and which contain a fixed locally Bass order (an order whose local pieces are all Bass orders, a mild singularity condition). On this class, the isogeny graph has a generalized volcano structure, with vertices organized into levels and a rim whose shape is read off from the order and its overorders. Earlier generalizations required restrictive additional assumptions such as maximal real multiplication, ordinary, and absolutely simple, while the theorem here also covers non-simple and non-ordinary isogeny classes. To get there, the paper first proves a structure theorem for the lattice of inclusions among overorders of a locally Bass order in an étale algebra, obtained by studying local singularities.

What carries the argument

The load-bearing mechanism is the ideal-theoretic dictionary: in the commutative-endomorphism-ring setting, isogenies from a variety correspond to invertible ideals of its endomorphism order. To use that dictionary in arbitrary dimension, the paper proves a structure theorem for the lattice of inclusions among the overorders of a locally Bass order in an étale algebra—a locally Bass order being one whose localization at every prime is a Bass order, so the singularities stay mild. The structure theorem is built from a careful study of those local singularities, and it yields the precise poset in which the overorders sit. That poset is what the isogeny graph mirrors, with levels of the volcano corresponding to ranks in the overorder lattice and edges corresponding to invertible ideals moving between neighboring overorders.

What would settle it

Compute the graph of isogenies of one fixed prime degree for a non-simple, non-ordinary abelian surface whose endomorphism order is locally Bass, and check whether the graph has the predicted volcano levels: a single rim at each level with the specified overorder relations. Finding two isogenous surfaces at the same claimed level that are connected by a prime-degree isogeny, or an overorder that appears in the graph but is missing from the predicted lattice of overorders, would break the structural claim.

Watch

Extended reading notes

Core claim

The central discovery is a structural theorem for isogeny graphs in arbitrary dimension: if $A$ is an abelian variety with commutative endomorphism ring $\operatorname{End}(A)$ and the endomorphism order contains a fixed locally Bass order, then the graph of varieties isogenous to $A$ is a generalized volcano. The graph's levels are governed by the overorders of the endomorphism order, and the edge structure is governed by the invertible ideals of those orders. This removes the restrictions of maximal real multiplication, ordinarity, and absolute simplicity that appeared in earlier work, so the result applies to non-simple and non-ordinary isogeny classes as well. The authors establish the graph theorem by first classifying the poset of overorders of a locally Bass order inside an étale algebra through a local analysis of singularities; the isogeny graph structure is a consequence of that classification together with the ideal-theoretic correspondence between isogenies and invertible ideals.

Load-bearing premise

The load-bearing premise is that every isogeny between varieties with commutative endomorphism ring is faithfully described by an invertible ideal of the associated order, and that this correspondence preserves the graph structure in every dimension, including the non-simple and non-ordinary cases.

Editorial extensions

If this is right

The classical one-dimensional volcano theorem now has a counterpart in every dimension for varieties with commutative endomorphism ring satisfying the locally Bass condition.
Non-simple and non-ordinary isogeny classes, which were outside the scope of earlier results, are covered by the same structural description.
The overorder lattice theorem provides an independent classification tool for the possible endomorphism orders of varieties in a fixed isogeny class.
The examples included in the paper show that these generalized volcanos can display phenomena not seen in ordinary elliptic-curve isogeny graphs.

Reading between the lines

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

Editorial extension: if the overorder poset is computable from local data at primes dividing the conductor, then building the full isogeny graph should reduce to computing those local pieces, which could be tested by enumerating small dimension-two examples.
Editorial extension: the inclusion of non-ordinary classes suggests the result may constrain which endomorphism orders can coexist in one isogeny class, since every vertex's order must appear in the same overorder lattice.
Editorial extension: in settings where these graphs are used for isogeny-based cryptography, the volcano structure would imply that path-finding moves up and down a hierarchy of levels, so the higher-dimensional graphs inherit the same search-problem shape that makes the one-dimensional case tractable.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Sarah, this one is worth a look but needs a careful read. Arpin–Marseglia–Springer claim a volcano structure theorem for isogeny graphs of abelian varieties of any dimension whose endomorphism ring is commutative and contains a fixed locally Bass order. That genuinely extends Brooks–Jetchev–Wesolowski, which required maximal real multiplication, ordinary, and absolutely simple. They also prove a structure theorem for the poset of overorders of a locally Bass order in an étale algebra, which sounds independently useful. The examples are a plus, and the authors are credible people in this area.

The soft spot is the ideal-theoretic dictionary. The whole theorem rests on matching isogenies to invertible ideals of the Bass order. On elliptic curves that dictionary is classical. In higher dimension, and especially for non-simple abelian varieties, it is not automatic. The stress-test example I keep coming back to: take A=E1×E2 with two non-isogenous ordinary CM elliptic curves, End(A)=O1×O2. A cyclic subgroup K=⟨(P,Q)⟩ is not necessarily stable under all of End(A). If some β in O2 moves Q off its cyclic subgroup, then the endomorphism (1,β) sends K to a different subgroup, so K cannot be the kernel A[I] of any invertible ideal I. If the paper's graph still counts that isogeny, the dictionary is incomplete; if it excludes it, the graph is only the O-linear part, not the full isogeny graph as advertised. The abstract gives no theorem statement that lets us check which.

That said, I don't see an internal contradiction in what's printed, and the issue may well be addressed in the body—maybe they define the graph in terms of O-linear isogenies, or restrict to subgroups stable under the order. I just can't tell from the abstract. This is not a demonstrated flaw; it's the key thing to verify in a full read.

Bottom line: if the proofs work, this is a significant result that will get cited. It's for people working on isogeny graphs, isogeny-based crypto, and the classification of isogeny classes in higher dimension. I'd send it to a serious referee who knows both orders and abelian varieties, because the stakes are high and the abstract cannot carry the burden. I wouldn't desk-reject it. For my own work, I'd wait until I can see the actual dictionary before citing it.

Referee Report

3 major / 3 minor

Summary. The paper (arXiv:2508.03570) announces graph structural theorems for isogeny graphs of abelian varieties of arbitrary dimension whose endomorphism ring contains a fixed locally Bass order. The authors claim a generalization of Kohel's volcano theorem for ordinary elliptic curves to non-simple and non-ordinary isogeny classes, going beyond the prior work of Brooks, Jetchev, and Wesolowski, which required maximal real multiplication, ordinary reduction, and absolute simplicity. The approach is said to be based on an ideal-theoretic dictionary between isogenies and invertible ideals of orders, and on a new structure theorem for the lattice of overorders of a locally Bass order in an étale algebra. Several examples of volcanoes and isogeny graphs with unexpected phenomena are promised. The full text was not supplied; this report is based on the abstract and the associated reviewer discussion.

Significance. If the announced results are correct, this would be a substantial and valuable generalization of one of the central structural results in computational arithmetic geometry, with likely applications to isogeny-based cryptography and to the study of abelian varieties over finite fields. The independent structure theorem for overorders of locally Bass orders also has potential interest beyond the isogeny-graph context. The paper aims at a wide class including non-simple and non-ordinary varieties, which is genuinely more delicate than the elliptic-curve case. However, the abstract alone provides no proof or proof sketch, so the significance is conditional on the correctness of the ideal-theoretic dictionary in the claimed generality.

major comments (3)

[Abstract] The central claim rests on an 'ideal-theoretic perspective on isogeny graphs,' but the paper does not state precisely which isogenies are edges of the graph. In the elliptic-curve setting, every isogeny between ordinary curves with a given order corresponds to an invertible ideal of that order. In higher dimension, especially for non-simple abelian varieties, not every isogeny has a kernel of the form A[I] for an invertible ideal I of the order O. For example, if A = E1 × E2 with End(A) = O1 × O2 and K = ⟨(P,Q)⟩ is a cyclic subgroup where P and Q have order ℓ and Q is not stable under O2, then the isogeny A → A/K cannot be represented as A[I] for any ideal I of O1 × O2, because every A[I] is stable under all of O. If such isogenies are included in the graph, the claimed dictionary is false; if they are excluded, the theorem describes only the O-linear subgraph, which is a strictly weaker statement. The manuscript must define the graph explicitly and either prove the ideal-theoretic bijection in full generality or restrict the statement to isogenies whose kernels are O-stable, with the volcano structure proved for that class.
[Abstract] The abstract announces theorems without any proof sketch. Given that the proof is claimed to generalize Kohel's result through an ideal-theoretic correspondence in a setting that includes non-simple and non-ordinary abelian varieties, the reader cannot check whether the standard elliptic-curve arguments actually extend. In particular, the local singularity analysis of Bass orders is mentioned but no indication is given of how it controls the graph structure in the non-ordinary and non-simple cases. A revised version must include at least a precise theorem statement with definitions and a proof outline; ideally the full proofs should be available for verification.
[Abstract] The phrase 'containing a fixed locally Bass order' is ambiguous. It could mean that O is a subring of End(A), or that O is a subring of End(A) ⊗ Q, and it is unclear whether the fixed order is part of the vertex data or merely a common subring of the endomorphism rings of all vertices. This ambiguity affects the statement of the volcano theorem, since the levels of a volcano are typically indexed by orders. The authors should state the precise objects of the category whose isogeny graph is being studied.

minor comments (3)

[Abstract] The abstract refers to 'Kohel' without a citation or reference to the specific result being generalized; the paper should include the relevant reference (Kohel's thesis) in the abstract or introduction.
[Abstract] The promised examples of 'volcanoes and isogeny graphs exhibiting unexpected properties' are not visible in the supplied text; the authors should ensure that these examples are included and clearly explained in the full manuscript.
[Abstract] Minor wording: 'singular ideals in orders' in the title is not defined in the abstract; the authors should briefly indicate what the 'singular ideals' are, since they seem to be central to the announced structure theorem.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular reasoning identified; the paper advances new structural theorems built on cited prior work and independent ideal-theoretic arguments.

full rationale

The paper's abstract announces new graph structural theorems for isogeny graphs of abelian varieties in arbitrary dimension, based on an ideal-theoretic perspective and a structure theorem for overorders of locally Bass orders. No derivation chain is visible in the provided text that would reduce a claimed prediction or theorem to its own inputs. The claimed results are presented as extensions of prior work by Kohel and by Brooks, Jetchev, and Wesolowski, not as reformulations of those results. The skeptical concern about whether the ideal-theoretic dictionary covers non-simple and non-ordinary isogeny classes is a correctness or completeness question about unstated assumptions, not a circularity: it does not assert that any equation or theorem is defined in terms of the target result, nor that a fitted parameter is renamed as a prediction. Since the provided full text is not available, there is no quoted passage that exhibits a self-definitional step, a fitted input called prediction, a load-bearing self-citation, an imported uniqueness theorem, a smuggled ansatz, or a renaming of a known result. Under the hard rule that circularity may only be claimed with quotable evidence of the specific reduction, the honest finding is no significant circularity.

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

Only the abstract was available. No fitted parameters appear in a pure mathematics paper; the listed axioms are background assumptions invoked by the abstract's described approach.

assumptions (2)

domain assumption The standard dictionary between isogenies and invertible ideals of orders in arithmetic geometry holds in the generalized setting of locally Bass orders in higher dimensions.
The proof leverages an ideal-theoretic perspective on isogeny graphs; if this correspondence breaks for non-simple/non-ordinary varieties, the main theorem fails.
standard math Standard background results in algebra and number theory, such as the theory of orders, etale algebras, and commutative algebra.
The abstract mentions reliance on structure of orders and local singularities; these are standard frameworks in the field.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Isogeny graphs of abelian varieties and singular ideals in orders." pith.science (2026). https://pith.science/paper/DWDWHR42

@misc{pith2026250803570,
  author       = {Pith},
  title        = {Pith review of: Isogeny graphs of abelian varieties and singular ideals in orders},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DWDWHR42}},
  note         = {Machine review of arXiv:2508.03570}
}

read the original abstract

Famously, Kohel proved that isogeny graphs of ordinary elliptic curves are beautifully structured objects, now called volcanos. We prove graph structural theorems for abelian varieties of any dimension with commutative endomorphism ring and containing a fixed locally Bass order, leveraging an ideal-theoretic perspective on isogeny graphs. This generalizes previous results, which relied on restrictive additional assumptions, such as maximal real multiplication, ordinary, and absolutely simple (Brooks, Jetchev, Wesolowski 2017). In particular, our work also applies to non-simple and non-ordinary isogeny classes. To obtain our results, we first prove a structure theorem for the lattice of inclusion of the overorders of a locally Bass order in an \'etale algebra which is of independent interest. This analysis builds on a careful study of local singularities of the orders. We include several examples of volcanoes and isogeny graphs exhibiting unexpected properties ultimately due to our more general setting.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

30 extracted references · 29 canonical work pages

[1]

On the ubiquity of G orenstein rings

Hyman Bass. On the ubiquity of G orenstein rings. Math. Z. , 82:8--28, 1963

work page 1963
[2]

Isogeny graphs of ordinary abelian varieties

Ernest Hunter Brooks, Dimitar Jetchev, and Benjamin Wesolowski. Isogeny graphs of ordinary abelian varieties. Res. Number Theory , 3:Paper No. 28, 38, 2017

work page 2017
[3]

Polarizations of abelian varieties over finite fields via canonical liftings

Jonas Bergstr\"om, Valentijn Karemaker, and Stefano Marseglia. Polarizations of abelian varieties over finite fields via canonical liftings. Int. Math. Res. Not. IMRN , (4):3194--3248, 2023

work page 2023
[4]

Abelian varieties over finite fields with commutative endomorphism algebra: theory and algorithms

Jonas Bergstr \"o m , Valentijn Karemaker , and Stefano Marseglia . Abelian varieties over finite fields with commutative endomorphism algebra: theory and algorithms . arXiv e-prints , page arXiv:2409.08865, September 2024

work page Pith review arXiv 2024
[5]

J. A. Buchmann and H. W. Lenstra, Jr. Approximating rings of integers in number fields. J. Th\' e or. Nombres Bordeaux , 6(2):221--260, 1994

work page 1994
[6]

Orbital integrals and Ideal class monoids for a Bass order

Sungmun Cho , Jungtaek Hong , and Yuchan Lee . Orbital integrals and Ideal class monoids for a Bass order . arXiv e-prints , page arXiv:2408.16199, August 2024

work page arXiv 2024
[7]

Categories of abelian varieties over finite fields, I : A belian varieties over F _p

Tommaso Giorgio Centeleghe and Jakob Stix. Categories of abelian varieties over finite fields, I : A belian varieties over F _p . Algebra Number Theory , 9(1):225--265, 2015

work page 2015
[8]

Vari\'et\'es ab\'eliennes ordinaires sur un corps fini

Pierre Deligne. Vari\'et\'es ab\'eliennes ordinaires sur un corps fini. Invent. Math. , 8:238--243, 1969

work page 1969

Show all 30 references

[9]

Galbraith

Christina Delfs and Steven D. Galbraith. Computing isogenies between supersingular elliptic curves over F _p . Des. Codes Cryptogr. , 78(2):425--440, 2016

2016
[10]

Isogeny volcanoes and the SEA algorithm

Mireille Fouquet and Fran c ois Morain. Isogeny volcanoes and the SEA algorithm. In Algorithmic number theory ( S ydney, 2002) , volume 2369 of Lecture Notes in Comput. Sci. , pages 276--291. Springer, Berlin, 2002

2002
[11]

Ferrand and J.-P

D. Ferrand and J.-P. Olivier. Homomorphisms minimaux d'anneaux. J. Algebra , 16:461--471, 1970

1970
[12]

Everett W. Howe. Principally polarized ordinary abelian varieties over finite fields. Trans. Amer. Math. Soc. , 347(7):2361--2401, 1995

1995
[13]

On the computation of overorders

Tommy Hofmann and Carlo Sircana. On the computation of overorders. Int. J. Number Theory , 16(4):857--879, 2020

2020
[14]

Isogeny graphs with maximal real multiplication

Sorina Ionica and Emmanuel Thom\' e . Isogeny graphs with maximal real multiplication. J. Number Theory , 207:385--422, 2020

2020
[15]

Jordan and Bjorn Poonen

Bruce W. Jordan and Bjorn Poonen. The analytic class number formula for 1-dimensional affine schemes. Bull. Lond. Math. Soc. , 52(5):793--806, 2020

2020
[16]

Products of CM elliptic curves

Ernst Kani. Products of CM elliptic curves. Collect. Math. , 62(3):297--339, 2011

2011
[17]

u rgen Kl \

Markus Kirschmer and J \"u rgen Kl \"u ners . Enumerating orders in number fields . arXiv e-prints , page arXiv:2411.08568, November 2024

2024 arXiv
[18]

Endomorphism rings of elliptic curves over finite fields

David Kohel. Endomorphism rings of elliptic curves over finite fields . PhD thesis, University of California, Berkely, 1996

1996
[19]

The L -functions and modular forms database

The LMFDB Collaboration . The L -functions and modular forms database. https://www.lmfdb.org, 2025. [Online; accessed 11 July 2025]

2025
[20]

Isogeny Graphs, Modular Polynomials, and Applications

Chloe Martindale. Isogeny Graphs, Modular Polynomials, and Applications . Doctoral dissertation, Universiteit Leiden and Universit\'e de Bordeaux, 2018

2018
[21]

Computing square-free polarized abelian varieties over finite fields

Stefano Marseglia. Computing square-free polarized abelian varieties over finite fields. Math. Comp. , 90(328):953--971, 2021

2021
[22]

Computing base extensions of ordinary abelian varieties over finite fields

Stefano Marseglia. Computing base extensions of ordinary abelian varieties over finite fields. Int. J. Number Theory , 18(9):1957--1974, 2022

1957
[23]

Cohen- M acaulay type of orders, generators and ideal classes

Stefano Marseglia. Cohen- M acaulay type of orders, generators and ideal classes. J. Algebra , 658:247--276, 2024

2024
[24]

Abelian varieties over finite fields and their groups of rational points

Stefano Marseglia and Caleb Springer. Abelian varieties over finite fields and their groups of rational points. Algebra Number Theory , 19(3):521--550, 2025

2025
[25]

Algebraic number theory , volume 322 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]

J\" u rgen Neukirch. Algebraic number theory , volume 322 of Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] . Springer-Verlag, Berlin, 1999. Translated from the 1992 German original and with a note by Norbert Schappacher, With a...

1999
[26]

Abhishek Oswal and Ananth N. Shankar. Almost ordinary abelian varieties over finite fields. J. Lond. Math. Soc. (2) , 101(3):923--937, 2020

2020
[27]

An algorithm for computing modular forms on 0 (N)

Arnold Pizer. An algorithm for computing modular forms on 0 (N) . J. Algebra , 64(2):340--390, 1980

1980
[28]

Isogeny volcanoes

Andrew Sutherland. Isogeny volcanoes. The Open Book Series , 1(1):507–530, November 2013

2013
[29]

Endomorphisms of abelian varieties over finite fields

John Tate. Endomorphisms of abelian varieties over finite fields. Invent. Math. , 2:134--144, 1966

1966
[30]

Waterhouse

William C. Waterhouse. Abelian varieties over finite fields. Ann. Sci. \' E cole Norm. Sup. (4) , 2:521--560, 1969

1969

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [1]

On the ubiquity of G orenstein rings

Hyman Bass. On the ubiquity of G orenstein rings. Math. Z. , 82:8--28, 1963

work page 1963

[2] [2]

Isogeny graphs of ordinary abelian varieties

Ernest Hunter Brooks, Dimitar Jetchev, and Benjamin Wesolowski. Isogeny graphs of ordinary abelian varieties. Res. Number Theory , 3:Paper No. 28, 38, 2017

work page 2017

[3] [3]

Polarizations of abelian varieties over finite fields via canonical liftings

Jonas Bergstr\"om, Valentijn Karemaker, and Stefano Marseglia. Polarizations of abelian varieties over finite fields via canonical liftings. Int. Math. Res. Not. IMRN , (4):3194--3248, 2023

work page 2023

[4] [4]

Abelian varieties over finite fields with commutative endomorphism algebra: theory and algorithms

Jonas Bergstr \"o m , Valentijn Karemaker , and Stefano Marseglia . Abelian varieties over finite fields with commutative endomorphism algebra: theory and algorithms . arXiv e-prints , page arXiv:2409.08865, September 2024

work page Pith review arXiv 2024

[5] [5]

J. A. Buchmann and H. W. Lenstra, Jr. Approximating rings of integers in number fields. J. Th\' e or. Nombres Bordeaux , 6(2):221--260, 1994

work page 1994

[6] [6]

Orbital integrals and Ideal class monoids for a Bass order

Sungmun Cho , Jungtaek Hong , and Yuchan Lee . Orbital integrals and Ideal class monoids for a Bass order . arXiv e-prints , page arXiv:2408.16199, August 2024

work page arXiv 2024

[7] [7]

Categories of abelian varieties over finite fields, I : A belian varieties over F _p

Tommaso Giorgio Centeleghe and Jakob Stix. Categories of abelian varieties over finite fields, I : A belian varieties over F _p . Algebra Number Theory , 9(1):225--265, 2015

work page 2015

[8] [8]

Vari\'et\'es ab\'eliennes ordinaires sur un corps fini

Pierre Deligne. Vari\'et\'es ab\'eliennes ordinaires sur un corps fini. Invent. Math. , 8:238--243, 1969

work page 1969