{"id":"6ff5021e-c149-40d2-8c32-a8fa2eb351c5","arxiv_id":"2608.02494","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Isogeny graphs of non-CM elliptic curves in characteristic 0 decompose into p-primary pieces, each isomorphic to one of the explicit trees H^r_{p^k}, H^r_{p∞}, or H^r_{p∞,+}, with r governed by a new p-blooming invariant.","lead":"This paper classifies every isogeny graph that can occur for elliptic curves without complex multiplication over fields of characteristic zero, showing each splits into prime-by-prime pieces of one of a few explicit shapes. It resolves a question previously settled only over the rational numbers, and ties each graph shape to the p-adic Galois representation of the curve.","discovery_kind":"first_principles","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Exhaustiveness of the p-primary classification depends on the unproved [70, Prop. 3.1] dichotomy; if that dichotomy has another branch, the families H^r_{p^k}, H^r_{p\\infty}, H^r_{p\\infty,+} are incomplete.","rationale":"The reader's verdict already identifies the same weakest assumption: exhaustiveness rests on [70, Prop. 3.1], which is not reproved in the manuscript. My stress-test confirms that this is the most load-bearing concern. Theorem 1 (primary decomposition) is fully proved and appears sound. The graph families and the p-blooming invariant are developed carefully, and the exceptional cases in Theorem 2 are consistent with the determinant/cyclotomic obstructions, which is evidence of care. The converse realizability passages are terse, but over characteristic 0 one can in principle realize prescribed open subgroups by base change to fixed fields or by generic points of modular curves, so that is a presentation issue rather than a mathematical gap. The external counting dichotomy, however, is the exact hinge on which 'no other p-primary graphs occur' depends. If [70] is correct and complete, the classification follows; if not, the central claim fails. This does not change the reader's CONDITIONAL verdict: the same condition should be discharged before the classification is regarded as unconditional.","tokens_in":72290,"tokens_out":20713,"duration_ms":235328,"concrete_test":"Audit [70, Prop. 3.1] at the level of its proof: extract from that proof the exact characterization of subgroups G <= B_0(p^k) for which the number of G-stable cyclic subgroups of order p^j is 2p^alpha, and verify that the criteria used in Theorem 4.25 to exclude this branch (the existence of a proper solution modulo p^{2r+1} and the factorization through E_{r+1}) are exactly the ones proved there, with no hidden hypothesis on K (e.g., roots of unity or unramifiedness). If this dichotomy cannot be reproduced from first principles, or if a third branch appears, Theorem 2's exhaustiveness is not established.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central classification (Theorem 2; finite case Theorem 4.25, infinite case Theorem 6.15) asserts that every p-primary isogeny graph of a non-CM elliptic curve over characteristic-0 K is one of H^r_{p^k}, H^r_{p\\infty}, or H^r_{p\\infty,+}. The decisive counting step in both forward directions is imported from the proof of Novak [70, Prop. 3.1]: for E0 as chosen in Theorem 4.25, the number of K-rational p^j-isogenies is stated to be either p^{min{alpha, floor(j/2)}} or 2p^alpha, with alpha = min v_p(a-c) over a Borel-conjugate image. This dichotomy is not proved in the manuscript; Proposition 2.2 only records the weaker finite list of possible counts, not the structural description in terms of alpha. The exclusion of the 2p^alpha branch in the proof of Theorem 4.25 relies on the internal description of that branch in [70] (the 'proper solution modulo p^{2r+1}' criterion), and Theorem 6.15 inherits the same dependency through Lemma 6.14. If [70] has any additional branch, or if the 2p^alpha branch can occur under a condition not ruled out by the paper's setup, then the families on the right side of Theorem 2 are not exhaustive and the central classification is incomplete. This is the single most load-bearing concern: it affects both the finite and infinite parts of the main theorem, and it is external to the otherwise fully developed Galois-representation machinery in Sections 4-6.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":72593,"tokens_out":4904,"duration_ms":59726,"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":[{"comment":"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","section":"§4.2, proof of Theorem 4.25"},{"comment":"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.","section":"§4.2 and §6.2, converses of Theorems 4.25 and 6.15"},{"comment":"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.","section":"§6.1, Definition 6.3 and the convention for H^∞_{p∞,+}"}],"minor_comments":[{"comment":"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.","section":"Abstract"},{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"§5, Proposition 5.10"},{"comment":"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.","section":"§7, Table 2"}],"recommendation":"major_revision","confidential_remarks":"The central structural result (Theorem 1) appears sound, and the classification itself is plausible and well developed. My recommendation is driven by the single load-bearing dependence on the proof details of Novak [70, Prop. 3.1] and by the under-specified Galois-existence step in the converses. Both are fixable within the manuscript’s scope: the authors could include a full proof of the counting dichotomy or clearly state it as a theorem with hypotheses and proof, and they could expand the converse realization argument. If those are provided, I would expect the paper to be publishable in a strong number-theory journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The headline result is real: a full classification of which isogeny graphs occur for non-CM elliptic curves over characteristic-0 fields, reducing the problem to p-primary graphs and then giving explicit families H^r_{p^k}, H^r_{p∞}, H^r_{p∞,+}. The primary decomposition (Theorem 1, weak Cartesian product) is proved cleanly and is self-contained. The p-blooming invariant is also a genuinely nice idea — it is defined directly from the p-adic Galois image, is shown to be an isogeny class invariant, and the formula r = min{I_p, floor(k/2)} is proven rather than assumed. The exceptional cases (H^0_{2^k}, the ζ_8 and sqrt(p*) conditions) are handled with care and are consistent with the determinant/cyclotomic obstructions, which suggests the authors really worked through the details.\n\nThe main soft spot is exactly what the stress-test flags: the exhaustiveness of the classification is imported from Novak [70, Prop. 3.1], specifically the dichotomy that the number of K-rational p^j-isogenies is either p^{min{α, floor(j/2)}} or 2p^α, with the structural description of the 2p^α branch. The paper only states the weaker per-degree count in Proposition 2.2, not the full dichotomy. If that dichotomy has another branch, or if the 2p^α exclusion is not fully justified, the graph families are incomplete. That is load-bearing for both the finite and infinite parts of Theorem 2. I would not call it a fatal flaw — the paper's internal logic is coherent and the dependency is on a recent result by one of the authors that is presumably correct — but it is the one point I would want a referee to verify, and ideally the paper should either include a proof of the dichotomy or make the precise statement of [70, Prop. 3.1] explicit and check that it applies verbatim.\n\nMinor points: the converse realization steps are terse in places (\"follows from the Galois correspondence\"), and I could not audit the computational appendices or the GitHub repository. Neither is disqualifying. The paper also recovers the rational classification and gives applications to potential CM and to an algorithm from the adelic image, which is a nice bonus.\n\nWho is this for? Number theorists working on isogeny graphs, Galois representations, or modular curves. It deserves a serious referee — I would send it out. The referee should be asked specifically to check the counting dichotomy and its application, and to confirm that no other branch can occur.","headline":"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.","tokens_in":73282,"tokens_out":1372,"would_cite":true,"duration_ms":18675,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G05","11G07","11G15","11F80","14K02","05C25","05C51"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["elliptic curves","isogeny graphs","Galois representations","p-adic representations","Cartesian product of graphs","p-blooming invariant","modular curves","potential complex multiplication"],"falsifier":"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.","tokens_in":72014,"feed_emoji":"🔀","tokens_out":2931,"duration_ms":35088,"temperature":0.7,"pith_summary":"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.","feed_headline":"All isogeny graphs of non-CM curves over char 0 now classified","feed_subtitle":"Each p-primary piece is a small explicit tree family; only one shape never occurs, and the rest are all realizable.","key_machinery":"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.","core_discovery":"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","pith_inferences":["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."],"forward_implications":["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."],"fun_headline_variants":["All char-0 non-CM isogeny graphs now classified","Non-CM isogeny graphs in char 0: complete classification","Char-0 isogeny graphs: every possible shape pinned down","Non-CM char-0 isogeny graphs: all but one shape realizable","Non-CM isogeny graphs over char 0 fully resolved"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["All char-0 non-CM isogeny graphs now classified","Non-CM isogeny graphs in char 0: complete classification","Char-0 isogeny graphs: every possible shape pinned down","Non-CM char-0 isogeny graphs: all but one shape realizable","Non-CM isogeny graphs over char 0 fully resolved"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000635,"raw_usage":{"total_tokens":2883,"prompt_tokens":979,"completion_tokens":1904,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":723,"completion_tokens_details":{"reasoning_tokens":1808}},"tokens_in":723,"tokens_out":1904,"duration_ms":19785,"temperature":1.0,"reasoning_tokens":1808,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-04T06:11:08.428379+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}