{"id":"14fec150-b66f-4e8d-a5af-5cec52892ff8","arxiv_id":"2506.06260","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The order of the elliptic fiber E_t in Kum(E1×E2) is exactly d(n)=n (odd n) or n/2 (even n), with one subtle CM exception, and every integer occurs as some order.","lead":"This paper computes the exact order of elliptic constant cycle curves that arise as fibers of a natural fibration on Kummer surfaces, establishing that the order is n or n/2 depending on the parity of n. As a consequence, every positive integer can be realized as the order of a constant cycle curve on some K3 surface.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Lemma 3.4's 'one can check' lattice-index assertion is load-bearing: Proposition 3.5's injectivity of pi^*_tr and hence ord(E_t)=ord(kappa'_t) depends on it. The exact equality in Theorem 1.1 is not secure until these indices are verified.","rationale":"The reader's weakest assumption pinpoints the same load-bearing point: the 'one can check' lattice-index assertion in Lemma 3.4. I agree that this is the most critical spot. The non-isogenous and 4-not-divide-n cases have independent Chow-theoretic arguments in Section 4.2, but they still use Proposition 3.5 to transfer the result from the abelian threefold back to the Kummer surface. The CM isogenous case additionally depends on Lemma 4.5's congruence computation; I regard that as a secondary numerical check, not the single most load-bearing step. The virtue of focusing on Lemma 3.4 is that it is a self-contained lattice-theoretic verification: no new Hodge theory is needed, and a positive check would substantially raise confidence in Theorem 1.1. The concern is not that the assertion is false; it is that the proof as written leaves a crucial index computation unstated, and the exact equality in the theorem is precisely what depends on it. No ad hominem is intended: this is an ordinary proof gap that an expert can close or refute. The verdict remains CONDITIONAL pending this check, so no change to the reader's verdict is needed.","tokens_in":17938,"tokens_out":9454,"duration_ms":94044,"concrete_test":"Use Nikulin's description of the Kummer lattice K and the standard lattice decomposition H^2(Bl(A),Z) congruent H^2(A,Z) direct-sum direct-sum Z[E_i] to compute the two inclusions in Lemma 3.4 explicitly: form the Gram matrices for K, direct-sum Z[E_i], and pi^*H^2(X,Z), then compute the Smith normal forms of the inclusions. Verify [H^2(Bl(A),Z):pi^*H^2(X,Z)] = [direct-sum Z[E_i]:pi^*(K)] = 2^11 and that the natural map direct-sum Z[E_i]/pi^*(K) -> H^2(Bl(A),Z)/pi^*H^2(X,Z) is an isomorphism. If the indices differ or the map has kernel, re-examine whether pi^*_tr is injective on the indicated 2-torsion classes; if injectivity fails, Proposition 3.5 collapses and the theorem's lower bound requires a different argument.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central reduction is Proposition 3.5, which identifies ord(E_t) with ord(kappa'_t) and uses Lemma 3.4. Lemma 3.4 claims pi^*_tr is injective, and its proof reduces to two unchecked lattice-theoretic statements: the embeddings pi^*H^2(X,Z) subset H^2(Bl(A),Z) and pi^*(K) subset direct-sum Z[E_i] both have index 2^11, and the resulting cokernel is identified with (direct-sum Z[E_i]/pi^*(K)) tensor H^1(E_t,Z). These are stated as 'one can check' with no computation or reference. If either index is wrong, or if the cokernel identification has a hidden kernel, then ker(pi^*) need not lie in the algebraic part J^3_alg(X times E_t); pi^*_tr could fail to be injective on torsion. In that case the equality ord(E_t)=ord(kappa'_t) would be replaced by an inequality, and the lower bound d(n) in Theorem 1.1 would not follow. The later computations of ord(kappa'_t) in Sections 4.2 and 4.3 are correct only conditional on this reduction, so the entire exact-order theorem hinges on this 'one can check' assertion.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the Huybrechts order of elliptic constant cycle curves E_t on Kummer surfaces X = Kum(E_1 × E_2) associated to torsion points t ∈ E_1 of order n > 2. The main theorem (Theorem 1.1) claims ord(E_t) = d(n) := n/2 if 2|n and n otherwise, with a single excluded case (E_1, E_2 non-isomorphic CM isogenous with 4|n). The proof uses a general strategy via the transcendental intermediate Jacobian J^3_tr(X × E_t) (Section 2.2), then reduces to an abelian threefold E_1 × E_2 × E_t: Lemma 3.1 constructs a d(n)-torsion compactification [Z'_t] of the Chow class κ'_t, Lemma 3.4 and Proposition 3.5 aim to show ord(E_t)=ord(κ'_t), and Sections 4.2–4.3 compute ord(κ'_t) using Chow group decompositions and the Abel–Jacobi map. Section 5 gives an example in the excluded case where the order is 1 or 2 depending on numerical conditions. The paper concludes that every positive integer occurs as the order of a smooth genus-one constant cycle curve.","tokens_in":18195,"tokens_out":31166,"duration_ms":273533,"significance":"If the main theorem is correct, this is a significant contribution: it gives the first exact computations of the order of non-rational constant cycle curves on K3 surfaces and shows that all orders can be realized. The strategy of using the transcendental intermediate Jacobian together with CTSS83 injectivity is natural and well executed in principle. The paper also provides an explicit example illustrating the subtle excluded CM case. The main result is concrete and falsifiable, and the corollaries are striking. However, several verification steps (the lattice index assertions in Lemma 3.4 and the congruence computation in Lemma 4.5) are not fully demonstrated, so the current version is not yet acceptable.","major_comments":[{"comment":"The proof of Lemma 3.4 contains two unchecked lattice-theoretic assertions that are load-bearing for Proposition 3.5. It is asserted that the embeddings π^*H^2(X,Z) ⊂ H^2(Bl(A),Z) and π^*(K) ⊂ ⊕ Z[E_i] both have index 2^11, and that these induce a natural isomorphism ⊕Z[E_i]/π^*(K) ≃ H^2(Bl(A),Z)/π^*H^2(X,Z), whence ker(π^*) ≃ (⊕Z[E_i]/π^*(K)) ⊗ H^1(E_t,Z). No computation or reference is supplied for these indices. If either index is incorrect, the identification of ker(π^*) with a subgroup lying in the algebraic part may fail, and the equality ord(E_t)=ord(κ'_t) in Proposition 3.5 would be replaced by an inequality, so the lower bound d(n) in Theorem 1.1 would not follow. Please provide a complete verification (e.g., a Gram-matrix computation for the Kummer lattice K and its pullback) or a precise reference that contains the computation.","section":"Section 3.3, Lemma 3.4"},{"comment":"In the proof of Lemma 4.5, the passage from equation (22) to the displayed system of congruence relations is not shown. The coefficients a_{ij} in the second congruence are never defined, and the reduction modulo 2H^3 requires a nontrivial expansion of the cycle representatives. Since Lemma 4.5 is the key input for Proposition 4.4 (the isogenous case, including the no-CM case), this omission is load-bearing. Please expand the computation in full or provide a lemma that states the congruence system with explicit definitions of all coefficients.","section":"Section 4.3, Lemma 4.5"},{"comment":"Remark 2.9 claims that the injectivity of the transcendental Abel–Jacobi map on torsion subgroups, proved in Lemma 2.6 for K3 surfaces, holds for any surface S with H^1(S,Z) ≠ 0 by 'the same argument'. The argument in Lemma 2.6 uses that Φ^alg_X is an isomorphism; for a surface with H^1 ≠ 0 (e.g., an abelian surface A), the map Φ^alg_S is only surjective with a nontrivial kernel, so the lifting of a torsion class to a torsion class in CH^1(S)⊗CH^1(C)_hom is not immediate. Since the injectivity of Φ^tr_A and Φ^tr_{Bl(A)} is used in Proposition 3.5, please provide a proof (e.g., using divisibility of the kernel) or a reference.","section":"Remark 2.9"}],"minor_comments":[{"comment":"The statement 'Any elliptic curve E can be embedded ... into any Kummer surface X = Kum(E × F)' is confusing, since the fibres of Kum(E×F) are isomorphic to F, not to E. If the intended meaning is that every elliptic curve appears as a fibre for a suitable choice of the Kummer surface, please rephrase.","section":"Corollary 1.2"},{"comment":"The word 'integeral' should be 'integral'.","section":"Definition 2.2"},{"comment":"The word 'repectively' should be 'respectively'.","section":"Section 5"},{"comment":"The displayed identity 'D⊗θ = 2D⊗α = π_*(π^*(D))⊗α = π^*_alg(D⊗α)' contains a type error: π^*(D) is not defined for D ∈ NS(Bl(A)); the correct pull-push relation is π^*(π_*(D))⊗α = π^*_alg(π_*D⊗α). As written, the reader cannot follow the argument.","section":"Proof of Lemma 3.4"},{"comment":"The expression '4d1([t] - [e1]) × [E1]' should be '4d1([t] - [e1]) × [E2]'.","section":"Proposition 4.3(b)"},{"comment":"The notation [id] is used both for the identity correspondence on E_2×E_2 and for the class in Hom(J(E_1),J(E_2)); please clarify the identification.","section":"Lemma 4.5"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is promising and, if the computational gaps are filled, likely publishable. The 'one can check' in Lemma 3.4 is the main concern; I would ask the author to include the lattice computation in an appendix. The paper fits the scope of the journal. No issues with attribution; the use of [Huy14] and [CTSS83] is appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"You should know two things about this paper. The main theorem is a genuine advance: it computes the exact order of elliptic constant cycle curves on Kummer surfaces, not just the upper bound ord(C) | n from Huybrechts, and it yields the first realization of every integer as an order. The method is also new: this is the first nontrivial use of the transcendental intermediate Jacobian route to pin down orders. The paper is well written and the author is honest about the excluded case, giving an explicit example where the order degenerates.\n\nThe soft spot is real. The reduction in Proposition 3.5, which identifies ord(E_t) with ord(kappa'_t), rests on Lemma 3.4. There the paper asserts without proof that two lattice embeddings have index 2^11 and that the cokernel identifies as claimed ('one can check'). That assertion is load-bearing: if either index is off or the cokernel has a hidden kernel, injectivity of pi^*_tr could fail and the lower bound d(n) would not follow. The later computations in Section 4 are conditional on this step. An expert in Kummer lattice theory could likely verify the indices in a few lines, but the paper as written does not include those lines. A referee should ask for them. The congruence computation in Lemma 4.5 is also compressed; it is probably correct but needs checking.\n\nI do not see circularity or invented entities. The cited [CTSS83] is the standard injectivity result, and Huybrechts's lemma is used as a definitional upper bound, not as this author's result. The example in Section 5 is a helpful sanity check rather than a handwave.\n\nWho is this for? People working on Chow groups of K3s and constant cycle curves. It deserves a serious referee: the result is consequential and the strategy is reusable, but the missing lattice verification should be supplied before publication. My own verdict would be conditional accept pending that check. I would bring it to a reading group as a good example of the intermediate Jacobian method, but I would flag Lemma 3.4.\n\nRecommendation: send to peer review. The gap is fillable and localized, and the payoff justifies the referee time.","headline":"First exact computation of constant cycle curve orders on Kummer surfaces, with a load-bearing 'one can check' lattice-index claim that a referee should verify.","tokens_in":18752,"tokens_out":2469,"would_cite":true,"duration_ms":23481,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14C25","14J28","14H52"],"pacs":[],"model":"deepseek-v4-flash","headline":"This paper proves that the order of elliptic constant cycle curves on Kummer surfaces is determined by the parity of the torsion order: it equals $n/2$ for even $n$ and $n$ for odd $n$, and every positive integer occurs.","keywords":["constant cycle curves","K3 surfaces","Kummer surfaces","Chow groups","intermediate Jacobians","Abel-Jacobi maps","elliptic curves","torsion points"],"falsifier":"Check the two indices in the proof of Lemma 3.4: the embeddings $\\pi^*H^2(X,\\mathbb{Z})\\subset H^2(\\mathrm{Bl}(A),\\mathbb{Z})$ and $\\pi^*(K)\\subset\\bigoplus\\mathbb{Z}[E_i]$ are claimed to have index $2^{11}$; computing either cokernel and finding a different order would break the identification $\\ker(\\pi^*)\\simeq(\\bigoplus\\mathbb{Z}[E_i]/\\pi^*(K))\\otimes H^1(E_t,\\mathbb{Z})$. Also, for a non-isomorphic isogenous CM pair with $4\\mid n$ outside $n=4$, computing $\\mathrm{ord}(E_t)$ would test whether the theorem's excluded case really is the only obstruction.","tokens_in":1869,"feed_emoji":"🔢","tokens_out":8174,"duration_ms":102564,"temperature":0.7,"pith_summary":"Constant cycle curves are curves whose points all represent the same class in the Chow group of the ambient K3 surface; the order measures how far the diagonal is from being decomposable, with rational curves being exactly the order-one case. This paper determines the order for the elliptic fibres that occur on Kummer surfaces built from a product of two elliptic curves: if the fibre lies over a torsion point of order $n>2$, the order is $n/2$ for even $n$ and $n$ for odd $n$, except in a narrow CM case. The result is the first exact computation of orders larger than one for non-rational constant cycle curves via intermediate Jacobians, and it implies that every positive integer is realized as the order of a smooth genus-one constant cycle curve on some K3 surface.","feed_headline":"Parity decides order of elliptic constant cycle curves","feed_subtitle":"For a torsion point of order n, the order is n/2 or n; every positive integer occurs.","key_machinery":"The computation rests on three objects: the torsion class $[Z'_t]=2([t]-[e_1])\\times[\\mathrm{id}]$ in $\\mathrm{CH}^2(E_1\\times E_2\\times E_t)$, the transcendental intermediate Jacobian $J^3_{\\mathrm{tr}}(X\\times E_t)$ (the part of the third intermediate Jacobian coming from the transcendental cohomology), and the theorem that the transcendental Abel–Jacobi map is injective on torsion classes. The class $[Z'_t]$ is a compactification of the torsion class $\\kappa'_t$ whose order equals $\\mathrm{ord}(E_t)$; its image in the intermediate Jacobian has order $d(n)$ because $[\\mathrm{id}]$ is a primitive class with self-intersection $-2$. A structural lemma on product cycles in triple products of elliptic curves then shows that no smaller multiple of $[Z'_t]$ can come from $\\mathrm{CH}^1(E_1\\times E_2)\\otimes\\mathrm{CH}^1(E_t)$, except in the excluded CM configuration.","core_discovery":"The central claim is Theorem 1.1: let $E_1,E_2$ be elliptic curves and $t$ a torsion point of order $n>2$; then the fibre $E_t$ in the Kummer surface $\\mathrm{Kum}(E_1\\times E_2)$ has order $d(n)$, equal to $n/2$ when $n$ is even and $n$ when $n$ is odd, except possibly when $E_1,E_2$ are non-isomorphic, isogenous with complex multiplication and $4\\mid n$. The proof constructs a torsion class $2([t]-[e_1])\\times[\\mathrm{id}]$ in the Chow group of the abelian threefold $E_1\\times E_2\\times E_t$, shows its order is $d(n)$ via the Abel–Jacobi map, and proves that this order is preserved when passing back to the Kummer surface. In the isogenous even case the preservation step uses a second-order test with the addition morphism and the intermediate Jacobian; the excluded case is genuinely exceptional, as an explicit example shows both order $1$ and order $2$ can occur for $n=4$ according to the lattice of the two CM curves.","pith_inferences":["The same intermediate-Jacobian strategy could be applied to other families of constant cycle curves on K3 surfaces, for example the ramification curves of double planes, where only divisibility bounds are currently known.","The parity dichotomy $n/2$ versus $n$ suggests the order may be governed by whether the involution $(-1)$ on the elliptic curve acts trivially on the relevant torsion class; a testable prediction is that any elliptic fibration fibre over a torsion section of order $n$ would satisfy the same dichotomy whenever the fibre has a smooth model on a K3 surface.","One could try to extend the explicit congruence calculation to all $4\\mid n$ CM cases and determine exactly when the order drops below $n/2$; the answer is likely a congruence condition on the endomorphism ring and the torsion point."],"forward_implications":["For every integer $n>1$, any elliptic curve can be embedded as a constant cycle curve of order exactly $n$ into some Kummer surface $\\mathrm{Kum}(E\\times F)$, so arbitrary orders occur in genus one.","Every positive integer is realized as the order of a smooth genus-one constant cycle curve on a K3 surface, sharpening the previous state where only order one and divisibility bounds were known.","When $4\\nmid n$ or the two elliptic curves are non-isogenous, the order is exactly $d(n)$, so the known upper bound $\\mathrm{ord}(E_t)\\mid n$ is improved by halving it for even $n$.","A concrete $n=4$ example with non-isomorphic isogenous CM curves shows the excluded case is not vacuous: the order can be $1$ or $2$ depending on divisibility conditions on the lattices."],"supporting_citations":[{"why":"Supplies the definition of the order of a constant cycle curve, the upper bound $\\mathrm{ord}(E_t)\\mid n$, and the strategy of computing orders via intermediate Jacobians.","marker":"[Huy14]"},{"why":"Proves the injectivity of Abel–Jacobi maps on codimension-two torsion classes, used to transfer order computations from Chow groups to intermediate Jacobians.","marker":"[CTSS83]"},{"why":"Provides the exact sequence relating $\\mathrm{CH}^2(S\\times C)$ to $\\mathrm{CH}^2(S\\times k(C))$, used to define the order and lift torsion classes.","marker":"[Blo10]"},{"why":"Supplies the injectivity of pullback on Chow groups under blow-up, used in the comparison between the Kummer surface and the abelian threefold.","marker":"[Ful98]"},{"why":"Gives the rank and discriminant of the Kummer lattice, used in the lattice-index computations of Lemma 3.4.","marker":"[BHPV03]"}],"fun_headline_variants":["Parity decides order of elliptic constant cycles","Elliptic curve order: n/2 for even n, n for odd n","Every integer is a constant cycle order on a K3","Torsion point order sets Kummer cycle order","Even torsion halves curve order on Kummer"],"cache_read_input_tokens":20864,"weakest_assumption_plain":"The step that carries the whole proof is that the order of the curve can be read off from the order of a torsion class on the abelian threefold; this relies on an injectivity statement for transcendental pullback maps whose proof contains a lattice-index computation asserted without details.","fun_headline_variants_meta":{"raw":{"variants":["Parity decides order of elliptic constant cycles","Elliptic curve order: n/2 for even n, n for odd n","Every integer is a constant cycle order on a K3","Torsion point order sets Kummer cycle order","Even torsion halves curve order on Kummer"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000507,"raw_usage":{"total_tokens":2448,"prompt_tokens":898,"completion_tokens":1550,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":514,"completion_tokens_details":{"reasoning_tokens":1471}},"tokens_in":514,"tokens_out":1550,"duration_ms":10594,"temperature":1.0,"reasoning_tokens":1471,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T05:57:50.210117+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the two indices in the proof of Lemma 3.4: the embeddings $\\pi^*H^2(X,\\mathbb{Z})\\subset H^2(\\mathrm{Bl}(A),\\mathbb{Z})$ and $\\pi^*(K)\\subset\\bigoplus\\mathbb{Z}[E_i]$ are claimed to have index $2^{11}$; computing either cokernel and finding a different order would break the identification $\\ker(\\pi^*)\\simeq(\\bigoplus\\mathbb{Z}[E_i]/\\pi^*(K))\\otimes H^1(E_t,\\mathbb{Z})$. Also, for a non-isomorphic isogenous CM pair with $4\\mid n$ outside $n=4$, computing $\\mathrm{ord}(E_t)$ would test whether the theorem's excluded case really is the only obstruction.","supporting_citations":[],"review_version":1}