{"id":"02cb62cc-69ff-4b62-a14e-25b160952282","arxiv_id":"2508.03570","paper_version":1,"verdict":"UNVERDICTED","confidence":"LOW","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":0,"one_line_summary":"Isogeny graphs of abelian varieties with locally Bass order have a generalized volcano structure in any dimension, covering non-simple and non-ordinary cases.","lead":"This paper proves that isogeny graphs of abelian varieties of any dimension with a commutative endomorphism ring have a volcano-like structure, generalizing a classical result for elliptic curves. The result covers previously excluded cases like non-simple and non-ordinary varieties, with potential consequences for isogeny-based cryptography.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The theorem's load-bearing step is the claimed ideal-theoretic dictionary between isogenies and invertible ideals of locally Bass orders; the abstract gives no proof that this dictionary covers non-simple, non-ordinary isogenies, where finite subgroup schemes need not be stable under the order.","rationale":"The reader's weakest assumption—that the ideal-theoretic dictionary between isogenies and invertible ideals is the load-bearing step—is exactly the concern I identify. The abstract explicitly attributes the proof method to this dictionary, and the novelty claim includes non-simple and non-ordinary cases where the classical elliptic-curve dictionary is least secure. My concrete test targets a simple non-simple configuration: a finite cyclic subgroup of a product of two ordinary CM elliptic curves that is not stable under the full endomorphism ring, hence cannot arise as the kernel attached to an order ideal. If the full proof handles such kernels by some additional argument, the concern is retired and the theorem may stand; if not, the graph analyzed is a proper subgraph of the isogeny graph, which would weaken the claimed generalization. No internal contradiction is demonstrated from the abstract alone, so the appropriate verdict remains unverified rather than accept or reject. The reader's UNVERDICTED verdict is therefore unchanged.","tokens_in":668,"tokens_out":12411,"duration_ms":167231,"concrete_test":"Locate the theorem or proposition in the full text that establishes the ideal-isogeny bijection, and test it on the following non-simple example. Let A=E1×E2 over a finite field, with Ei ordinary CM elliptic curves having non-isogenous, commutative endomorphism ring O1×O2. Choose a prime ℓ and points P∈E1[ℓ], Q∈E2[ℓ] such that some β∈O2 satisfies β(Q)∉⟨Q⟩. Let K=⟨(P,Q)⟩ and consider the isogeny A→A/K. The kernel K is not stable under the endomorphism (1,β), so K cannot be A[I] for any invertible ideal I of O1×O2. Trace the paper's dictionary on this isogeny: if it cannot associate an invertible ideal to K, the dictionary excludes a genuine isogeny; if it excludes the edge from the graph, the theorem describes only the O-linear subgraph rather than the full isogeny graph claimed in the abstract.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central structural claim—a volcano theorem for isogeny graphs of abelian varieties with commutative endomorphism ring containing a fixed locally Bass order—stands or falls on the ideal-theoretic dictionary announced in the abstract. To prove the graph theorem, the authors must establish a bijection between isogenies A→B in the relevant class and invertible ideals of a locally Bass order O⊂End(A), and must show this bijection is compatible with the graph in arbitrary dimension, including non-ordinary and non-simple cases. This is precisely where the elliptic-curve argument does not automatically generalize. For an abelian variety A=E1×E2 with non-isogenous ordinary CM elliptic factors, End(A)=O1×O2 is commutative. A finite cyclic subgroup K=⟨(P,Q)⟩ with P,Q of order ℓ need not be stable under End(A): if some β∈O2 satisfies β(Q)∉⟨Q⟩, then the endomorphism (1,β) maps K to a different subgroup. Such K cannot equal A[I]=∩_{α∈I} ker α for any invertible ideal I of O1×O2, because every ker α is stable under all of O. If the paper's dictionary nevertheless counts the isogeny A→A/K, it is incomplete; if it excludes this edge, the graph described is at most the O-linear subgraph, not the full isogeny graph in the newly claimed non-simple, non-ordinary setting. The abstract provides no theorem statement or proof from which the validity of this dictionary can be checked, so this is the key unverified assumption rather than a demonstrated error.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":1010,"tokens_out":4590,"duration_ms":54186,"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":[{"comment":"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.","section":"Abstract"},{"comment":"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.","section":"Abstract"},{"comment":"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.","section":"Abstract"}],"minor_comments":[{"comment":"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.","section":"Abstract"},{"comment":"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.","section":"Abstract"},{"comment":"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.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The provided manuscript is effectively an abstract with no body, so a full technical assessment is impossible. The main reviewer concern about the ideal-theoretic dictionary is legitimate and must be addressed head-on: either the graph definition excludes non-O-linear isogenies, in which case the novelty should be framed accordingly, or the dictionary must be proved in a setting where the standard argument does not automatically apply. Given the broad claims, I would not recommend acceptance until the full proof is available and the ambiguity about the graph is resolved. If the full text already addresses these points, the authors should indicate the relevant sections in their revision response."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThat 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.\n\nBottom 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.","headline":"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.","tokens_in":1498,"tokens_out":3950,"would_cite":false,"duration_ms":39312,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G10","14K02"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["isogeny graphs","volcano structure","abelian varieties","commutative endomorphism rings","locally Bass orders","overorder lattices","étale algebras","non-ordinary isogeny classes"],"falsifier":"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.","tokens_in":496,"feed_emoji":"🌋","tokens_out":8491,"duration_ms":89788,"temperature":0.7,"pith_summary":"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.","feed_headline":"Volcano structure proven for higher-dimensional isogeny graphs","feed_subtitle":"Covers non-simple and non-ordinary abelian varieties, going beyond the classic elliptic-curve case.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[],"fun_headline_variants":["Isogeny graphs of abelian varieties are generalized volcanoes","Volcano structure for isogeny graphs in any dimension","Higher-dimensional isogeny graphs are volcanoes","From elliptic to abelian: isogeny graphs as volcanoes","Generalized volcano theorem for isogeny graphs"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Isogeny graphs of abelian varieties are generalized volcanoes","Volcano structure for isogeny graphs in any dimension","Higher-dimensional isogeny graphs are volcanoes","From elliptic to abelian: isogeny graphs as volcanoes","Generalized volcano theorem for isogeny graphs"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000294,"raw_usage":{"total_tokens":1692,"prompt_tokens":905,"completion_tokens":787,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":521,"completion_tokens_details":{"reasoning_tokens":710}},"tokens_in":521,"tokens_out":787,"duration_ms":6954,"temperature":1.0,"reasoning_tokens":710,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T04:20:02.322358+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}