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Algebraic cycles and values of Green's functions -- Products of Elliptic Curves

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2502.04608 v2 pith:LGJZPG3N submitted 2025-02-07 math.AG math.NT

Ramesh Sreekantan (with an appendix by Kannappan Sampath) This is my paper · ORCID
classification math.AGmath.NT MSC 11G1511G5514K2214C2514G3519E15
keywords motiviccycleshigherChowgroupsGreen'sfunctionscomplexmultiplicationKummerK3surfacesregulatorsweaklyholomorphicmodularformsBorcherdslifts
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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

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

  • 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.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

4 major / 6 minor

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.

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 (4)
  1. [§5.7, Theorem 5.6] 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.
  2. [§4.4, Theorem 4.6] 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.
  3. [§4.3–§4.4, Corollary 4.7] 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.
  4. [Appendix §7.2] 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.
minor comments (6)
  1. [Abstract and §1] 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.
  2. [§5.3] There is a typo in the name 'Brunier-Ehelen-Yang'; it should be 'Bruinier-Ehlen-Yang'.
  3. [§4.3] 'For evey node c' is a typo for 'For every node c'.
  4. [§5.7, Conjecture 5.9] 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.
  5. [§6.1] 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.
  6. [Appendix, beginning] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circular derivation: the algebraicity step is conditional on an unproved geometric identification, not on its own conclusion.

full rationale

The central construction in Section 4 is self-contained: the curve C_{n,1} is produced from Theorem 4.3 (Coray) and the boundary computation in Theorem 4.6 is carried out directly, with the nonzero coefficient a_phi computed from intersection data. The indecomposability conclusion uses Proposition 3.2, whose proof is independent. The regulator comparison Proposition 5.5 is built on Zhang's Proposition 5.2 and Soulé's Green-current formalism, not on the paper's own target. Theorem 5.6 is a general statement: it assumes a cycle with boundary sum a_tau S_tau and derives algebraicity from the regulator pairing; it does not assume the algebraicity conclusion. The only delicate step is Section 5.7, where the paper asserts that the boundary Ctilde_phi - Ctilde_-phi (or Gamma_phi - Gamma_-phi) is a multiple of the CM cycle at points of X cap T_n^gamma ('the cycle Gamma_phi - Gamma_-phi is (a multiple of) the CM cycle for those products of elliptic curves lying on the modular curve'), and then says the precise regulator formula 'seems reasonable to expect' and places it as Conjecture 5.9. This is a missing proof or a conditional step, but it is not circular: nothing in the construction of xi or in Theorem 5.6 assumes that identification; the algebraicity statement would follow if that identification were established. The appendix attempts a similar justification via orthogonality ('This implies that, on X_n3, the cycle ~Z_phi2 is a multiple of the CM cycle at a point on X_n2 cap X_n3'), again as a geometric argument rather than a restatement of the target. There is no fitted parameter disguised as a prediction, no self-citation used as the sole justification for a central premise, and no definition that builds the conclusion into the input. The self-citations to [Sre01], [Sre14], [Sre22], [Sre23], [Sre24a], and [Sre24b] are programmatic or auxiliary; the load-bearing external inputs are Zhang, Soulé, Esnault-Viehweg, Coray, and Sato.

Assumptions & free parameters 2 free parameters · 7 assumptions · 2 invented entities

No parameter was fitted to data; the odd degree n and the CM cycle normalization c are parameters of the construction, not empirical fits. The main burden falls on external theorems of Zhang, Esnault-Viehweg and Soulé, and on several unproved identification steps in Sections 4.4 and 5.7. The invented objects are mathematical cycles and a conjectural correspondence, with no external falsifiable handle beyond the conjectures themselves.

free parameters (2)
  • n (isogeny degree) = any odd positive integer
    Parameter of the construction, not fitted to data. The motivic cycles and their boundary depend on n, and the paper treats n as an arbitrary odd integer.
  • c (CM cycle normalization) = chosen so that (S_tau,S_tau) = -1
    Introduced in Section 5.4 to fix the cycle class eta_y. It is a normalization constant, not a fitted parameter, and is intended to drop out of regulator comparisons.
assumptions (7)
  • domain assumption Beilinson regulator is computed by Green's currents: for ∂(ξ)=∑ a_τ Z_τ, <reg_R(ξ|_y), ω_y> = ∑ a_τ ∫_{A_τ} ω_y g_{Z_τ}.
    Invoked in Section 5.5 in the proof of Proposition 5.5, citing Soulé Ch. III Theorem 1. It is the bridge from algebraic cycles to Green's functions.
  • domain assumption Zhang's Proposition 3.4.1: the Green's current of the CM cycle evaluated on eta_y equals (1/2)G^X_2(tau,y).
    Used as Proposition 5.2. This external theorem is what turns regulators into weight-2 Green's function values.
  • domain assumption Compatibility of the motivic intersection product and the Deligne cohomology cup product (Esnault-Viehweg, Proposition 7.4).
    Used in Theorem 5.6 to identify the regulator of Xi_y ∩ S_y with the logarithm of an algebraic number.
  • domain assumption Theorem 4.3: for odd n, existence and uniqueness of a rational curve of bidegree (n,1) meeting the eight-line configuration in 2n+1 specified points and tangencies.
    The proof is a dimension count showing non-emptiness, but uniqueness is asserted. This theorem is used in Proposition 4.4 to define C_{n,1} for all (lambda_1,lambda_2).
  • domain assumption Fibres over X ∩ T^γ_n have Picard number 4 and give CM points.
    Stated in Section 5.7 without proof. Needed so the boundary C̃_φ−C̃_−φ can be identified with CM cycles and Proposition 5.5 applies.
  • ad hoc to paper In Theorem 4.6, the function μ with f^ι_c = μ/f_c is identically 1 after normalizing f_c(s)=1.
    The normalization fixes μ(s)=1 at one moving section, not μ everywhere. The conclusion μ≡1 is needed for the computed boundary ∂(ξ_c)=a_φ(C̃_φ−C̃_−φ), so this is a gap in the proof.
  • ad hoc to paper The assumption that P^1_c lies on C̃_φ over T^γ_n affects the sign of the boundary.
    Acknowledged after Theorem 4.6. The sign is needed for the claimed identification of the boundary with a multiple of C̃_φ−C̃_−φ.
invented entities (2)
  • Motivic cycles ξ^{φ,γ}_{n,c}
    purpose: Constructed to link regulators of motives to higher Green's function values at CM points.
    These are new mathematical objects defined via Proposition 3.1 on Kummer K3 surfaces. The paper itself notes their non-triviality is not immediately clear, and their independence is asserted without a full regulator computation.
  • Conjectural motivic cycle ξ_f for a weakly holomorphic modular form f
    purpose: Postulated in Conjecture 6.1 to give a motivic interpretation of weakly holomorphic modular forms via the theta lift Phi_j(y,f).
    Its existence is exactly the conjecture; no independent construction or verification is provided.

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Pith. "Pith review of Algebraic cycles and values of Green's functions -- Products of Elliptic Curves." pith.science (2026). https://pith.science/paper/LGJZPG3N

@misc{pith2026250204608,
  author       = {Pith},
  title        = {Pith review of: Algebraic cycles and values of Green's functions -- Products of Elliptic Curves},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LGJZPG3N}},
  note         = {Machine review of arXiv:2502.04608}
}
read the original abstract

Gross and Zagier defined certain `higher Green's functions' on products of modular curves and conjectured that the value of these functions at complex multiplication points should be logarithms of algebraic numbers. This is now a theorem of Li and Bruinier-Li-Yang. We relate this conjecture to the existence of motivic cycles in the universal family of products of elliptic curves along the lines of Mellit and Zhang. Using this we are able to prove Zagier's conjecture in some cases when the two CM points have the same discriminant. This is originally a theorem of Viazovska. Li, Bruinier-Li-Yang, Bruinier-Ehlen-Yang, Viazovska and others relate this conjecture to Borcherds' lifts of weakly holomorphic modular forms. Their works, coupled with ours, suggest that there should be a link between motivic cycles in the universal family on the one hand and Borcherds lifts on the other. We explain why this is the case. This suggests a motivic interpretation of weakly holomorphic modular forms. In the special case we look at we show that indeed this is true.

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  1. A note on higher Green's functions

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