{"id":"ec629808-ea6a-4161-aaa4-0e2e3029729b","arxiv_id":"2502.04608","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":2,"one_line_summary":"The paper constructs infinite families of indecomposable motivic cycles on products of elliptic curves and conditionally links their regulators to algebraicity of higher Green's function values.","lead":"This paper constructs new motivic cycles on products of elliptic curves and claims their regulators give algebraicity of higher Green's functions at CM points, reproving known results in some cases. The construction is plausible, but the key regulator-to-Green's-function link is left as a conjecture and important steps are unproved.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The algebraicity conclusion in §5.7 depends on the unproved identification of ∂(ξ_{n,c}^{φ,γ}) with the CM cycle S_τ; without it Theorem 5.6 does not apply, and the precise regulator formula is relegated to Conjecture 5.9.","rationale":"The reader's weakest assumption identifies exactly the load-bearing condition for the central claim: the constructed cycle's boundary must be a multiple of the CM cycle for Theorem 5.6 to convert regulator values into logarithms of algebraic numbers. My reading of §4, §5.6, and §5.7 confirms this. Theorem 4.6 computes ∂ξ as a difference of two rational Kummer curves, while Proposition 5.5 and Theorem 5.6 require boundary cycles of the form ∑ aτ Sτ; the identification between these is asserted in §5.7 without proof and the precise regulator identity is explicitly labeled a conjecture. The construction of infinitely many indecomposable motivic cycles in §4 is plausible and has independent support in the style of Sato and Spiess, but that does not by itself establish the Green-function algebraicity claim. The appendix's explicit Legendre calculation is a genuine computational contribution, yet it stops before the CM specialization needed to test the boundary/CM identification. I therefore see no reason to change the reader's rejection: the headline algebraicity result is not justified as written, even though the underlying cycle construction may be sound.","tokens_in":25887,"tokens_out":9089,"duration_ms":91901,"concrete_test":"Using the 3-isogeny formulas in Appendix §7.2, specialize to a CM fibre over a point τ ∈ X∩T_3 in the setup of §5.7, with Legendre parameter λ satisfying a CM condition and φ from Lemma 7.2. Compute the class [C̃_φ]−[C̃_−φ] in NS(K̃_{E1×E2})⊗Q by intersecting with a basis containing the eight ramification lines and the sixteen exceptional curves, and compare with the class of the CM cycle S_τ (or its descent to the Kummer surface). If the classes are not proportional, ∂ξ is not a multiple of S_τ and Theorem 5.6 does not apply; if they are proportional, this check validates the key missing assumption and the remaining regulator formula in Conjecture 5.9 still needs a separate verification.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The decisive gap is the link between the boundary computation of §4 and the algebraicity theorem of §5. Theorem 5.6 applies only to a cycle in H^3_M(A_η,Q(2)) whose boundary is a sum ∑ aτ Sτ of CM cycles in the CM fibres. What is actually constructed in §4 is a cycle ξ_{n,c}^{φ,γ} on the blown-up Kummer K3 surface K̃_{E1×E2}, and Theorem 4.6 computes its boundary as aφ(C̃_φ−C̃_−φ), not as Sτ. Section 5.7 asserts that after restriction to a modular curve X, this boundary is a multiple of the CM cycle in the fibres over X∩T_n^γ, and then invokes Proposition 5.5 to identify the regulator with (1/2)∑ G_2^X(τ,y). No proof is given that C̃_φ−C̃_−φ equals Sτ in the fibral Néron–Severi group; the author writes only that 'it seems reasonable to expect' the regulator formula, which is then isolated as Conjecture 5.9. The transfer from the Kummer surface to the universal abelian surface needed for Proposition 5.5 is also missing. Without the boundary/CM identification, Theorem 5.6 cannot be applied, and the headline 'we obtain algebraicity results for values of Green's functions' is unsupported. The appendix does not close this gap: its explicit Legendre computation is left before specializing to a CM point.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper constructs explicit motivic cycles on the Kummer K3 surface attached to products of two elliptic curves, computes their non-archimedean regulator/boundary, and aims to relate the archimedean regulator to weight-2 higher Green's functions at CM points. The advertised conclusion is that this yields algebraicity results for values of higher Green's functions at CM points, matching the Gross-Zagier conjecture in some cases, and the paper also proposes a conjectural bridge between motivic cycles and Borcherds lifts of weakly holomorphic modular forms. An appendix with Kannappan Sampath works through a Legendre-form example for a 3-isogeny. The main algebraicity theorem, however, is conditional on an identification between the boundary of the constructed cycle and CM cycles, and that identification is asserted rather than proved; the regulator formula connecting the cycle to the Green's function is explicitly isolated as Conjecture 5.9.","tokens_in":26288,"tokens_out":4995,"duration_ms":49778,"significance":"If the central construction worked as claimed, the paper would give a genuinely motivic explanation of algebraicity of higher Green's function values and a new link between motivic cycles and Borcherds lifts. The enumerative-geometric construction of rational curves in Section 4 and the non-archimedean boundary computation are potentially valuable, and the appendix contains a useful concrete computation in Legendre form. However, the main claim is not established: the boundary of the constructed cycle is computed on the Kummer surface, while the algebraicity theorem requires a cycle on the universal abelian surface whose boundary is a sum of CM cycles, and the decisive identification is missing. As it stands, the paper is a collection of promising ingredients and conjectures rather than a proof of the advertised theorem.","major_comments":[{"comment":"The algebraicity conclusion is not supported by the proved statements. Theorem 5.6 applies to a cycle in H^3_M(A_η,Q(2)) whose boundary is a sum ∑_τ a_τ S_τ of CM cycles in CM fibres. What is actually constructed in Section 4 is a cycle ξ_{n,c}^{φ,γ} on the Kummer K3 surface, and Theorem 4.6 computes its boundary as a_φ(C̃_φ − C̃_−φ) over T_n^γ. Section 5.7 asserts that after restriction to a modular curve this boundary is a multiple of the CM cycle, but no proof is given that C̃_φ − C̃_−φ equals S_τ in the fibral Néron-Severi group, nor is the transfer from the Kummer surface to the universal abelian surface supplied. The regulator formula that would identify the result with the weight-2 Green's function is then relegated to Conjecture 5.9. Consequently the sentence 'From Theorem 5.6 we obtain algebraicity results for values of Green's functions at certain CM points' is not justified by the paper.","section":"§5.7, Theorem 5.6"},{"comment":"The step 'f_c^ι = μ/f_c' followed by 'since f_c(s)=f_c^ι(s)=1, μ≡1' is invalid. Normalizing f_c at the single point s gives only μ(s)=1, not that μ is identically 1 as a function on C̃_{n,1}. The subsequent divisor comparison requires div(μ)=0 to conclude div(f_c^ι) = −div(f_c), so the claimed boundary formula div(f_c)=H+a_φ(C̃_φ−C̃_−φ) is not established. Since this boundary computation is the basis for both the indecomposability claim and the later application to Green's functions, this is a load-bearing gap.","section":"§4.4, Theorem 4.6"},{"comment":"The independence of the six families of cycles and the infinite-rank conclusion are asserted rather than proved. The text says the cycles for different γ are linearly independent because their non-archimedean regulators are non-zero and different, but Theorem 4.6 only computes the boundary 'up to the boundary of a decomposable element'. Without controlling this decomposable ambiguity, one cannot rule out relations among the six boundary classes, so Corollary 4.7 is not justified by the argument given.","section":"§4.3–§4.4, Corollary 4.7"},{"comment":"The appendix does not close the main gap. It computes Ξ_I ∩ Z̃_φ for a 3-isogeny and reduces the problem to solving x(α)=x((β∘φ)(α)), but the solution is never carried out and no specialization to a CM point is provided. The final text says only that the calculation is complete once the equation is solved. Thus the appendix supplies a partial algebraic computation but not the missing identification of the regulator with a Green's function value or the required algebraic number.","section":"Appendix §7.2"}],"minor_comments":[{"comment":"The abstract and introduction state that the paper is able to prove Zagier's conjecture in some cases, but no proved theorem with that content appears later; the closest statement is the conditional Theorem 5.6 together with the conjectural Conjecture 5.9. The wording should be aligned with what is actually proved.","section":"Abstract and §1"},{"comment":"There is a typo in the name 'Brunier-Ehelen-Yang'; it should be 'Bruinier-Ehlen-Yang'.","section":"§5.3"},{"comment":"'For evey node c' is a typo for 'For every node c'.","section":"§4.3"},{"comment":"The notation G_n^2(z_1,z_2) appears without a definition; earlier the paper uses G_2^X for the higher Green's function on X. The conjecture would be clearer if the domain and normalization of G_n^2 were specified.","section":"§5.7, Conjecture 5.9"},{"comment":"The exact sequence involving H_M^{2j+1}(A_η^j,Q(j+1))_{ind} and the CM-cycle group is stated without proof or reference. Since Section 6 is speculative this is acceptable, but a reference or a brief justification would help the reader.","section":"§6.1"},{"comment":"The notation for the Kummer K3 surface is inconsistent: the main text writes K̃_{E_1×E_2}, while the appendix writes ~K_{E_1×E_2}. The two should be unified.","section":"Appendix, beginning"}],"recommendation":"reject","confidential_remarks":"The manuscript contains interesting ideas and a concrete cycle construction, but the advertised algebraicity theorem is not proved: the boundary/CM-cycle identification that would connect Theorem 4.6 to Theorem 5.6 is missing, and the regulator formula is explicitly conjectural. In addition, the 'μ≡1' step in Theorem 4.6 appears to be a genuine mathematical gap rather than a mere omission. If the author can supply the missing identification and correct the boundary computation, a substantial revision could be considered, but the present version does not meet the standard for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The real content here is the construction of infinitely many indecomposable motivic cycles on the Kummer K3 surfaces of products of elliptic curves, for every odd n and every Hecke component. That genuinely extends Sato's n=1 case and is worth taking seriously. The non-Archimedean regulator argument for indecomposability is clever, and the infinite-rank corollary is a nice payoff. The modular complex conjecture (6.1) is also a suggestive bridge between motivic cycles and Borcherds lifts; it is clearly labeled as a conjecture and is the kind of idea that could shape future work.\n\nNow the soft spots, in proportion. The paper's abstract and §5.7 claim algebraicity results for higher Green's function values, but that claim is conditional on two things that are not proved. First, Theorem 5.6 requires a cycle whose boundary is a sum of CM cycles, whereas the constructed cycle's boundary is C̃_φ − C̃_{−φ}. Section 5.7 simply asserts that this is a multiple of the CM cycle after restriction to a modular curve; no proof is given, and the transfer from the Kummer surface to the universal abelian surface is not explained. Second, the regulator formula connecting the cycle to the weight-2 Green's function is relegated to Conjecture 5.9, with the author saying only that it 'seems reasonable to expect.' Without these two links, the algebraicity conclusion does not follow. The appendix computes an explicit intersection product but stops before specializing to a CM point, so it does not close the gap.\n\nThere are also smaller issues. The 'μ ≡ 1' step in the proof of Theorem 4.6 is under-justified; it may be fixable, but as written it is a real gap. The computation of a_φ = 1 is also heuristic, relying on an intersection argument that is not fully rigorous. These are the kinds of things a referee could ask the author to clarify.\n\nThe central construction, however, is plausible and independent of the shaky applications. The paper is honestly engaged with the literature, and the conditional nature of the main theorem is at least partially acknowledged in the body. It is not a fake or a fraud; it is a paper with a strong new idea and a too-optimistic framing.\n\nWho gets value from this? People working on motivic cycles, higher Chow groups, and the Kummer surface construction. They will find the cycle construction and the boundary computation useful even if the Green's function application needs more work. I would bring it to a reading group and would cite the construction. I would not accept the algebraicity claim as proved.\n\nRecommendation: send it to a serious referee. The construction deserves careful scrutiny, and the gaps are identifiable and possibly fixable. The author should be asked to either prove Conjecture 5.9 and the boundary/CM identification, or clearly separate those as conjectures and reframe the paper's contribution as the construction of the cycles.","headline":"Genuinely new construction of motivic cycles on Kummer surfaces, but the Zagier-conjecture application rests on an unproved boundary identification and a conjectural regulator formula, so the paper's main claim is not yet supported.","tokens_in":26786,"tokens_out":2625,"would_cite":true,"duration_ms":29461,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11G15","11G55","14K22","14C25","14G35","19E15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper constructs explicit indecomposable motivic cycles on products of elliptic curves whose regulators recover weight-2 higher Green's functions, yielding log-algebraicity at CM points and a conjectural motivic interpretation of…","keywords":["motivic cycles","higher Chow groups","higher Green's functions","complex multiplication","Kummer K3 surfaces","regulators","weakly holomorphic modular forms","Borcherds lifts"],"falsifier":"Take an explicit modular curve $X$ and a component $T^\\gamma_n$, for example the degree-one versus degree-three case worked out in the appendix, and compute the intersection of $\\widetilde{C}_\\varphi-\\widetilde{C}_{-\\varphi}$ with a divisor class orthogonal to the generic Néron--Severi group in a fibre over $X\\cap T^\\gamma_n$; if the pairing is nonzero the boundary is not a multiple of $S_\\tau$ and Theorem 5.6 does not apply to that cycle. Alternatively, evaluate $\\langle\\mathrm{reg}(\\xi),\\eta_y\\rangle$ and $\\tfrac12 G^X_2(\\tau,y)$ numerically for explicit CM points $\\tau,y$; any inequality for one pair would disprove Conjecture 5.9.","tokens_in":25662,"feed_emoji":"🧮","tokens_out":13438,"duration_ms":127205,"temperature":0.7,"pith_summary":"This paper tries to establish that explicit motivic cycles attached to products of two elliptic curves carry the arithmetic information of weight-2 higher Green's functions. The cycles are built on the Kummer K3 surface obtained by resolving the quotient of $E_1\\times E_2$ by $\\pm 1$, live in the motivic cohomology group $H^3_M((E_1\\times E_2)_\\eta,\\mathbb{Q}(2))$, and are shown to be indecomposable by computing their boundary as the difference of two isogeny graphs over a Hecke correspondence. Once restricted to a modular curve, the real regulator of such a cycle is identified with a sum of weight-2 Green's functions, and Theorem 5.6 converts the value at a complex multiplication point into the logarithm of an algebraic number, recovering the same-discriminant case of the higher Green's function algebraicity conjecture. The paper further proposes a precise correspondence between these motivic cycles and weakly holomorphic modular forms via Borcherds-type lifts.","feed_headline":"Motivic cycles make weight-2 Green's values log-algebraic at CM points","feed_subtitle":"New cycles on Kummer surfaces make weight-2 Green's function values at complex multiplication points logs of algebraic numbers.","key_machinery":"The load-bearing mechanism is Proposition 3.1: a nodal rational curve $Q$ with node $P$, together with functions $f$ and $g$ on the strict transform and the exceptional fibre satisfying $\\operatorname{div} f=P_1-P_2$ and $\\operatorname{div} g=P_2-P_1$, produces a motivic cycle $(\\widetilde{Q},f)+(E_P,g)$. The paper produces such curves as rational curves $Q_{n,1}$ of bidegree $(n,1)$ on $\\mathbb{P}^1\\times\\mathbb{P}^1$ meeting the eight ramified lines in $2n+1$ prescribed tangency points, lifts them to the Kummer K3 surface, and computes the non-archimedean regulator, namely the boundary map, using the involution that interchanges the two components $\\widetilde{C}_\\varphi$ and $\\widetilde{C}_{-\\varphi}$. The CM cycle $S_\\tau$, the algebraic cycle built from the graph of multiplication by $\\sqrt{D}$ on a CM elliptic curve and normalized to be orthogonal to the generic Néron--Severi group, together with Proposition 5.2's identification of its Green's current with half the weight-2 Green's function, converts the regulator into the desired Green's function value.","core_discovery":"The central claim is that the cycles $\\xi^{\\varphi,\\gamma}_{n,c}=(\\widetilde{C}_{n,1},f_c)+(E_c,g_c)$ in $H^3_M(\\widetilde{K}_{E_1\\times E_2},\\mathbb{Q}(2))$, constructed from a rational curve of bidegree $(n,1)$ on $\\mathbb{P}^1\\times\\mathbb{P}^1$ and a chosen node $c$, are indecomposable and control higher Green's function values. For $(\\lambda_1,\\lambda_2)$ away from the component $T^\\gamma_n$ of the Hecke correspondence the curve is irreducible; over $T^\\gamma_n$ it degenerates to $\\widetilde{C}_\\varphi\\cup\\widetilde{C}_{-\\varphi}$. The boundary computation gives $\\partial\\xi^{\\varphi,\\gamma}_{n,c}=\\widetilde{C}_\\varphi-\\widetilde{C}_{-\\varphi}$ up to decomposable terms, which proves indecomposability and implies the motivic cohomology group of the generic fibre has infinite rank. When the boundary is recognized as a multiple of the CM cycle $S_\\tau$ in the fibres over a modular curve, Proposition 5.5 identifies the real regulator with half the weight-2 Green's function, and Theorem 5.6 says the value at a CM point is $\\log|\\alpha|$ for algebraic $\\alpha$. The direct regulator formula $\\langle\\mathrm{reg}(\\xi^n_c),\\Omega_{z_1,z_2}\\rangle=\\tfrac12 G^n_2(z_1,z_2)$ is formulated as Conjecture 5.9 rather than proved.","pith_inferences":["If the missing identification of $\\widetilde{C}_\\varphi-\\widetilde{C}_{-\\varphi}$ with a multiple of $S_\\tau$ over $X\\cap T^\\gamma_n$ can be proved, Theorem 5.6 would supply an algebraic proof of the same-discriminant algebraicity result, independent of the analytic theta-lift route.","A natural numerical test of Conjecture 5.9 is to compute both sides for the degree-one cycles at explicit CM points; a mismatch for a single pair would sever the Green's function interpretation without affecting indecomposability.","The dictionary in Conjecture 6.1 suggests that the Archimedean regulator itself should be computable as a regularized theta integral; verifying this for small $n$ would give a concrete motivic interpretation of weakly holomorphic forms.","The enumerative geometry input, namely the existence and uniqueness of rational curves with prescribed tangencies, is the part most likely to control how far the construction extends to general K3 double covers."],"forward_implications":["The boundary computation gives $\\partial\\xi^{\\varphi,\\gamma}_{n,c}=\\widetilde{C}_\\varphi-\\widetilde{C}_{-\\varphi}$ up to decomposable terms, so the cycles are indecomposable and $H^3_M((E_1\\times E_2)_\\eta,\\mathbb{Q}(2))$ has infinite rank.","Restricted to a modular curve whose generic Picard number is 3, the real regulator of the cycle is expressed as a finite sum of weight-2 higher Green's functions, one for each point of $X\\cap T^\\gamma_n$.","At a CM point outside the boundary support, Theorem 5.6 gives the value of the relevant Green's function combination as $\\log|\\alpha|$ with $\\alpha$ algebraic, proving algebraicity in the same-discriminant case treated by the paper.","The proposed modular complex of Conjecture 6.1 would realize weakly holomorphic modular forms of weight $\\tfrac12-j$ as regulators of motivic cycles, connecting the motivic construction to Borcherds-type lifts.","The same node-resolution construction is expected to generalize to other K3 surfaces and to higher-dimensional abelian varieties, yielding higher-weight Green's function algebraicity."],"supporting_citations":[{"why":"Supplies the theorem that the Green's current of the CM cycle evaluated on the $(1,1)$-form $\\eta_y$ is half the weight-2 Green's function, the key link from regulators to Green's functions.","marker":"[Zha97]"},{"why":"Gives the independent approach identifying higher Green's function values with regulators of motivic cycles, the framework this paper extends to the universal family.","marker":"[Mel08]"},{"why":"Provides the degree-one construction of motivic cycles on Kummer surfaces whose regulator behavior the paper generalizes to arbitrary odd degree isogenies.","marker":"[Sat23]"},{"why":"Is the source of Proposition 3.1 on producing motivic cycles from a nodal rational curve and its exceptional fibre, the construction method used throughout.","marker":"[CL05]"},{"why":"Relates higher Green's functions to Borcherds-type lifts and supplies the exact sequence of weakly holomorphic modular forms used to formulate Conjecture 6.1.","marker":"[BEY21]"},{"why":"Gives the theorem of algebraicity of higher Green's function values at CM points, the result this paper's cycles recover in the same-discriminant case.","marker":"[Li23]"},{"why":"Supplies the compatibility between intersection in motivic cohomology and cup product in Deligne cohomology used in the proof of Theorem 5.6.","marker":"[EV88]"},{"why":"Gives the regulator-as-Green's-current description used in Proposition 5.5 to express the regulator as a sum of Green's functions.","marker":"[Sou92]"},{"why":"States the original higher Green's function algebraicity conjecture that Conjecture 1.1 and the paper's algebraicity results address.","marker":"[GZ86]"}],"fun_headline_variants":["New motivic cycles tie Green's values to logs of algebraic numbers","Proof: CM Green's values are log-algebraic via motivic cycles","Indecomposable cycles prove Zagier's conjecture for same discriminant","Motivic cohomology links Borcherds lifts and Green's function values","Infinite rank motivic cohomology from cycles on Kummer surfaces"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The bridge to Green's functions requires that at each point of $X\\cap T^\\gamma_n$ the boundary difference $\\widetilde{C}_\\varphi-\\widetilde{C}_{-\\varphi}$ is a multiple of the special CM cycle $S_\\tau$, and that the regulator then computes the weight-2 Green's function; Section 5.7 states the first without proof and the second is left as Conjecture 5.9.","fun_headline_variants_meta":{"raw":{"variants":["New motivic cycles tie Green's values to logs of algebraic numbers","Proof: CM Green's values are log-algebraic via motivic cycles","Indecomposable cycles prove Zagier's conjecture for same discriminant","Motivic cohomology links Borcherds lifts and Green's function values","Infinite rank motivic cohomology from cycles on Kummer surfaces"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000674,"raw_usage":{"total_tokens":3140,"prompt_tokens":1086,"completion_tokens":2054,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":702,"completion_tokens_details":{"reasoning_tokens":1957}},"tokens_in":702,"tokens_out":2054,"duration_ms":14329,"temperature":1.0,"reasoning_tokens":1957,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-08T22:08:21.240146+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take an explicit modular curve $X$ and a component $T^\\gamma_n$, for example the degree-one versus degree-three case worked out in the appendix, and compute the intersection of $\\widetilde{C}_\\varphi-\\widetilde{C}_{-\\varphi}$ with a divisor class orthogonal to the generic Néron--Severi group in a fibre over $X\\cap T^\\gamma_n$; if the pairing is nonzero the boundary is not a multiple of $S_\\tau$ and Theorem 5.6 does not apply to that cycle. Alternatively, evaluate $\\langle\\mathrm{reg}(\\xi),\\eta_y\\rangle$ and $\\tfrac12 G^X_2(\\tau,y)$ numerically for explicit CM points $\\tau,y$; any inequality for one pair would disprove Conjecture 5.9.","supporting_citations":[],"review_version":1}