$p$-adic Higher Green's Functions for Stark-Heegner Cycles

Hazem Hassan

REVIEW 3 major objections 4 minor 28 references

$p$-adic Higher Green's Functions for Stark-Heegner Cycles

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper constructs a higher-weight p-adic Green's pairing between real-quadratic divisors and conjectures that principal values are p-adic logarithms of algebraic numbers in predicted abelian extensions.

desk verdict A genuinely new higher-weight construction with attractive numerics, but the existence proof for J_{k,D} skips the hard case (Deg != 0) and the numerics never test it. read the letter →

arxiv 2509.09446 v2 pith:E3RTUKRB submitted 2025-09-11 math.NT

Hazem Hassan This is my paper · ORCID

classification math.NT MSC 11F6711F8511G40

keywords rigidmeromorphiccocycleshigherGreen'sfunctionsp-adicupperhalf-planeRM-divisorsrealquadraticfieldsmodularsymbolsBruhat-Titstreeclass-fieldlogarithms

topics Birch and Swinnerton-Dyer

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 extends the weight-2 theory of rigid meromorphic cocycles to higher even weights, producing a p-adic higher Green's pairing G_k(D1,D2) between real-quadratic divisors on the p-adic upper half-plane. For each inert RM-divisor D, the author builds a canonical weight-k cocycle J_{k,D} whose divisor is supported on the Γ-orbits of D, and pairs it with a second divisor through a boundary duality. The central conjecture is that when D1 is principal, G_k(D1,σ) is a linear combination of p-adic logarithms of algebraic numbers in the compositum of the narrow class fields attached to the two divisors. The paper reports a computer implementation and numerical evidence, including p=3, k=4 calculations matching the conjecture to over 140 digits of p-adic accuracy. If correct, this gives the real-quadratic analogue of higher Green's functions and indirect evidence for the existence of algebraic analogues of Heegner cycles in the real-quadratic setting.

What carries the argument

The load-bearing object is the boundary-measure modular symbol Φ_{k,D}, defined on edges of the Bruhat-Tits tree of the unramified quadratic extension of Q_p: on edges outside the Q_p-tree it records, for each RM-point w in the Γ-orbit, the intersection number of the geodesics (w,w̄) and (r,s) times the degree-(k-2) polynomial in T determined by w. This function is almost harmonic: its divergence is the degree modular symbol Deg_{k,D}. Adding a correction χ on the Q_p-tree makes it harmonic, after which a distribution-theoretic lift from bounded harmonic measures to rigid analytic functions produces a function on the unramified extension, and an eventual-stabilization step extends it to a ri

What would settle it

Run the paper's algorithm for a new principal RM-divisor, say at p=5, k=4, and compare the first 150 5-adic digits of J^0_{k,D1}[σ] against the predicted Q-linear combination of p-adic logarithms in the stated compositum; any disagreement would refute Conjecture 5.11. Alternatively, compute the corrected boundary measure Φ_{k,D}+χ on the tree of the unramified quadratic extension for some strong-degree-zero D and check whether it satisfies the stabilization condition of Lemma 2.24; a single infinite path with non-zero stable value would block the existence of J_{k,D} as a meromorphic cocycle.

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Extended reading notes

Core claim

The paper's central discovery is a canonical construction, for any inert RM-divisor D of strong degree zero, of a higher-weight rigid meromorphic cocycle J_{k,D} whose divisor is the prescribed divisor Div_{k,D} supported on Γ-orbits of real-quadratic points. The Green's function is defined as the period pairing G_k(D1,D2)=J_{k,D1}[y^#_{k,D2}] between this cocycle and a canonically chosen homology class attached to D2. The paper proves the resulting pairing recovers the weight-two logarithm of the multiplicative cocycle when k=2, and that for k>2 the construction is governed by the classical correspondence between modular symbols and p-new eigenforms. It then conjectures that for principal D

Load-bearing premise

The construction depends on the eventual-stabilization condition (Lemma 2.24): the corrected boundary measure on the Bruhat-Tits tree of the unramified quadratic extension must be bounded and must stabilize to zero along all but finitely many infinite paths, so that the lifted function extends from rigid analytic on the quadratic extension to genuinely meromorphic on H_p; the paper asserts this for the corrected measures Φ_{k,D}+χ rather than proving it in full generality.

Editorial extensions

If this is right

At k=2 the new pairing reduces to log_p of the known real-quadratic singular moduli, so the higher-weight theory is a direct generalization and can be probed against existing computations.
For principal divisors, the conjecture predicts explicit algebraic numbers in specific class fields, turning the Green's values into a source of computable p-adic logarithms; the p=3,k=4 examples already realize this.
The k>2 cohomology is spanned by p-new eigenforms, so the higher Green's functions carry modular-form information and could appear in p-adic special-value formulas for L-functions of the relevant higher-dimensional cycles.
The conjectural prime-factorization formula would give a purely quaternionic description of the q-adic valuations of the algebraic numbers, linking RM-values to optimal embeddings and geodesic intersection numbers.
If algebraicity holds, the values provide an analytic construction of elements of narrow class fields of real quadratic fields, complementing the CM case where singular moduli generate class fields.

Reading between the lines

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

One could test the conjecture beyond the examples by computing the obstruction Obs_L for small discriminants and using it to search systematically for non-trivial principal combinations; the paper gives the framework but does not carry out such a search.
The same boundary-measure recipe might extend to ramified RM-points or to composite-level congruence subgroups; the author indicates the ramified case is deferred to a thesis, and if the stabilization lemma holds there, the range of testable examples would broaden considerably.
If the conjecture is established, the resulting logarithms could be assembled into a p-adic regulator on the relevant higher-dimensional cycles, giving a concrete shadow of the conjectural real-quadratic Heegner cycles; this is an implication the paper leaves implicit.
The prime-factorization conjecture suggests a reciprocity law for these p-adic logarithms: the q-adic valuations are zero unless q is inert in both orders, and otherwise equal weighted intersection numbers; this could be verified computationally by factoring the algebraic numbers in the numerical examples to high precision.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

I've read the paper carefully. My take: the higher weight generalization of Darmon-Vonk is real and the numerical algebraicity is striking, but the main existence theorem has a load-bearing gap that the paper does not acknowledge, and the computations avoid it entirely.

What's new: the construction of J_{k,D} as ST(Phi_{k,D}+chi) and the higher Green's pairing G_k(D1,D2) are original, and the explicit examples (e.g., equations (21) and (23)) give high-precision evidence for Conjecture 1.1. The paper also correctly positions itself relative to Negrini's cocycles and Rotger-Seveso's Stark-Heegner cycle conjecture. That's worth a serious look.

The soft spot is Theorem 3.30(2). The proof says: find chi with grad(chi) = -Div_{k,D}, using the surjectivity of the co-restriction map; then ST(Phi_{k,D}+chi) is a rigid analytic function, and Lemma 2.24 gives the meromorphic extension. But two things are never checked: (i) the co-restriction map's surjectivity is asserted, not proved; (ii) even granting it, there is no argument that chi can be chosen bounded (so ST applies), nor that Phi+chi satisfies the eventual stabilization hypothesis of Lemma 2.24. For D with Deg_{k,D}=0, chi=0 and everything works; for Deg_{k,D} != 0, the entire harmonicity correction comes from chi, and no proof is given. The numerics in Section 5.2 only use divisors with Deg_{4,D1} trivial, so they never test this case. That's not a minor technicality: G_k is defined for all inert RM-divisors, but the existence of J_{k,D} for the nontrivial case is the one the paper needs.

I don't think this is a fatal flaw. The conjectures are framed conditionally, and the construction is plausible on its face. But the paper as written overstates what is proved. I would send it to a serious referee, with a request to focus on exactly this gap--either supply the missing boundedness/stabilization argument or state the needed condition as an assumption. As it stands, it's a promising preprint, not an accepted result.

Referee Report

3 major / 4 minor

Summary. The paper proposes a higher-weight generalization of Darmon–Vonk's rigid meromorphic cocycles for the Ihara group Γ=SL₂(Z[1/p]). For an inert RM-divisor D, the author constructs a boundary function Φ_{k,D} on the Bruhat–Tits tree of the unramified quadratic extension K/Q_p, and then, after adding a correction χ to make it harmonic, applies the Schneider–Teitelbaum lift and Lemma 2.24 to obtain a weight-k rigid meromorphic cocycle J_{k,D} with prescribed divisor. The higher Green's function is defined by evaluating J_{k,D₁} on a homology class y^♯_{k,D₂} attached to a second RM-divisor (Definition 4.22). Conjectures 1.1 and 5.11 assert that these values are linear combinations of p-adic logarithms of algebraic numbers in the compositum of the relevant narrow class fields. Numerical examples for p=3, k=4, with deg-zero divisors D₁, exhibit such logarithms to high p-adic precision.

Significance. If the construction is completed, this gives a new real-quadratic analogue of higher Green's functions and Stark–Heegner cycles, with explicit algebraicity predictions and a conjectural factorization formula. The paper is commendably concrete: it provides an algorithm (Algorithm 5.7), reports high-precision numerical evidence, and does not fit parameters to obtain the alleged algebraic numbers; the α_i are outputs of the computation. The claimed link to Negrini's weight-k cocycles and to Rotger–Seveso's Darmon cycles is a natural and potentially important step. However, the load-bearing existence theorem for J_{k,D} is not fully proved, and the numerical evidence intentionally avoids the difficult case. The central claim is therefore conditional in its present form.

major comments (3)

[§3.4, Theorem 3.30(2) and Lemma 2.24] The existence of J_{k,D} for arbitrary inert RM-divisors D is asserted but not established. The proof of Theorem 3.30(2) reduces the problem to finding χ∈MS_Γ(C_p¹(P_n)) with ∇χ = −Deg_{k,D}, says ∇ is surjective by identifying it with a co-restriction map, and then applies ST to Φ_{k,D}+χ. Three load-bearing points are missing. First, no proof or reference is given that Deg_{k,D} lies in the image of the co-restriction map from MS_{Γ_0(p)}(P_n) to (MS_{SL₂(Z)}(P_n))²; the statement 'which is known to be surjective' is not enough. Second, even if a χ exists, the Schneider–Teitelbaum lift applies only to bounded harmonic functions; no boundedness of χ or of Φ_{k,D}+χ is shown. Third, Lemma 2.24 requires an eventual stabilization hypothesis on all paths in P(v,K), whereas Lemma 3.20 only establishes finiteness of Σ∩U_e for each fixed edge e. Since Φ_{k,D} vanishes on T_p by construction, f
[§5.2, Examples 5.9 and 5.10] The numerical evidence only exercises the case Deg_{k,D}=0, for which χ=0 and the difficult part of Theorem 3.30(2) disappears. Both examples state 'Deg_{4,D₁} is trivial' and use the direct ST lift of Φ_{k,D}. Thus they do not test the existence of a bounded, stabilizing correction χ, which is precisely the gap in the central construction. Since the main conjecture is formulated for 'principal' divisors (Obs₀(J_{k,D})=0), and the paper leaves open whether such divisors can have Deg_{k,D}≠0, the evidence is consistent with a version of the theory restricted to strong degree zero, but it does not support the full theorem as stated.
[§4.2, Definition 4.22 and Lemma 4.6] Lemma 4.6 defines the canonical cocycle J_{k,D} by subtracting a bounded analytic cohomology class from an arbitrary lift J′_{k,D} with the correct divisor. This relies on the identifications in Proposition 3.10, but the proof of that proposition invokes Lemma 3.9 and Corollary 3.7, which in turn use the Hecke-module isomorphism of Corollary 3.7 only for k>2. For k=2 the argument is not covered, and for k>2 it is still contingent on the boundedness asserted in Proposition 3.4. The manuscript should make explicit that Lemma 4.6 is conditional on the missing parts of Theorem 3.30(2). As written, the uniqueness claim 'there exists a unique cohomology class' is too strong relative to what is proved.

minor comments (4)

[§3.4, proof of Theorem 3.30(2)] In the second paragraph of the proof, the symbols are interchanged: the text writes '∇χ{r,s}=−Div_{k,D}{r,s}' and then 'Div_{k,D} is in the image of MS_Γ(C_p¹(P_n))→MS_Γ(C_p⁰(P_n))'. Since Div_{k,D} takes values in divisors, not in Cₚ⁰(P_n), the condition should involve Deg_{k,D}. This is confusing and should be corrected.
[Lemma 2.24] The hypothesis reads 'for every vertex v∈T_p¹', but T_p¹ is the set of edges; the intended object is presumably a vertex of T_p or a path beginning at a vertex of T_p. Also, 'eventually stabilizes to 0' is weaker than what the proof uses: it needs stabilization to the prescribed residue polynomials P_j at the finitely many support points of the exceptional paths. The statement should be aligned with the proof.
[Notation, Definition 4.11] The object y_{k,σ}=γ_σ⊗(σ)⊗((T−σ)^{n/2}(T−σ̄)^{n/2}/(σ−σ̄)^{n/2}) is called a 'Stark–Heegner cycle', but it is a homology class in H₁(Γ, Div(H_p)⊗P_n), not an algebraic cycle. The terminology is evocative but should be flagged as conjectural, especially since the introduction distinguishes the open problem of constructing Stark–Heegner cycles.
[§5.1, Algorithm 5.7] The text says the code is available on the author's website. For reproducibility, an archival version with version-control metadata and a precise list of SageMath dependencies would be preferable. This is not a scientific issue, but it would help readers verify the 145-digit claims.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the higher Green's function values are computed outputs, not fitted inputs; the principal-divisor hypothesis is not the same as the algebraicity conclusion.

full rationale

The central derivation defines Φ_{k,D} directly from the RM-divisor D (Definition 3.23), lifts via the Schneider–Teitelbaum isomorphism when Deg_{k,D}=0 (Definition 3.27), and then defines the Green's function by G_k(D1,D2)=J_{k,D1}[y^♯_{k,D2}] (Definition 4.22). The algebraic numbers α_i in Conjecture 1.1 and Examples 5.9–5.10 are outputs of the computation, not parameters fitted to match the right-hand sides: Algorithm 5.7 computes J^L_{k,D1}{0,∞} recursively and evaluates it at σ, with no fitting step. The 'principal' assumption (Obs_0(J_{k,D}) trivial, Definition 4.1) guarantees existence of a Coleman primitive J^L, which makes the appearance of logarithms in the notation natural, but the substantive claim that the resulting infinite logarithmic expression collapses to a finite linear combination of logarithms of algebraic numbers in the compositum of narrow class fields is not encoded in that assumption. No load-bearing step is justified by a self-citation: the cited results [DV21, Neg23, RS12, dS09] are published external/independent works, and Lemma 4.6's uniqueness is proven in the paper. The unproved boundedness/stabilization assertion in Theorem 3.30(2) — the existence of χ with ∇χ=−Deg_{k,D} and the eventual stabilization hypothesis of Lemma 2.24 — is a genuine correctness gap that the paper itself does not discharge; the numerics deliberately use only Deg_{4,D}=0, where χ=0. But this is a gap or proof-risk, not a circular reduction: the theorem does not assume J_{k,D} exists in order to construct J_{k,D}. Accordingly the circularity score is 0.

Assumptions & free parameters 1 free parameters · 5 assumptions · 2 invented entities

The central construction rests on standard p-adic analysis and cohomology tools, but also on two ad hoc assumptions specific to this paper: surjectivity of the co-restriction map used to solve the divergence equation, and the unverified stabilization condition of Lemma 2.24. No continuous free parameters are fitted to force the algebraicity results; the only parameter is the logarithm branch L, and its dependence is part of the conjecture.

free parameters (1)

Branch of p-adic logarithm log_L = L = 0 in the computations
The Green's function requires a choice of Coleman primitive of the meromorphic cocycle. The computations use log_0, and Conjecture 5.11 asserts that the dependence on L is exactly through log_L(α_i). This is a normalization, not a number fitted to the target values.

assumptions (5)

standard math Morita and Breuil duality pairings and the Schneider-Teitelbaum lift are valid for the relevant weight k modules, including the integral and bounded variants.
These are established theorems cited from Morita, Breuil, de Shalit, and Schneider-Teitelbaum, used throughout Sections 2 and 3.
standard math Eichler-Shimura isomorphism and the decomposition H^1(Γ0(p), P_n) ≅ S_k ⊕ S_k for p-new forms, and the vanishing of H^2(Γ0(1), P_n).
Invoked in Theorem 3.6, Corollary 3.7, and Lemmas 3.8-3.10 to identify modular symbols, cohomology, and obstruction maps.
ad hoc to paper The co-restriction map MS_Γ(C^1_p(P_n)) → MS_Γ(C^0_p(P_n)) is surjective and is identified with the map to (P_n^{SL2(Z)})^2.
Asserted in the proof of Theorem 3.30(2) as 'known to be surjective' without a proof or a specific reference. This surjectivity is needed to solve ∇χ = -Div_{k,D}.
ad hoc to paper The harmonic function Φ_{k,D}+χ satisfies the eventual stabilization hypothesis of Lemma 2.24, so its Schneider-Teitelbaum lift extends to a meromorphic function on H_p.
The hypothesis is stated in Lemma 2.24, but the paper does not verify it for the constructed Φ_{k,D}+χ. This is load-bearing for the existence of J_{k,D} in M(k;K).
domain assumption All RM-points considered are inert in the associated real quadratic field; ramified points are deferred to the author's thesis.
The paper restricts to inert RM points in Definition 3.11 and Remark 3.31, so the classification and Green's function are not claimed for ramified points.

invented entities (2)

p-adic higher Green's function G_k(D1,D2)
purpose: Defined as the RM-period of the higher weight meromorphic cocycle J_{k,D1} against the unitary divisor y^♯_{k,D2}; conjectured to encode logarithms of algebraic numbers and to be the real quadratic analogue of the archimedean height pairing of Heegner cycles.
The object is new to this paper. Its main support is the numerical evidence in Section 5, which is internal to the paper and not an external falsifiable prediction.
Stark-Heegner cycle homology class y_{k,σ} = γ_σ⊗(σ)⊗((T-σ)^{n/2}(T-σ̄)^{n/2}/(σ-σ̄)^{n/2})
purpose: A cohomological avatar of the conjectural algebraic Stark-Heegner cycles; the Green's function is evaluated on these homology classes.
The existence of actual algebraic cycles is conjectural and not established; the paper only constructs a homology class in H_1(Γ, Div(H_p)⊗P_n).

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Pith. "Pith review of $p$-adic Higher Green's Functions for Stark-Heegner Cycles." pith.science (2026). https://pith.science/paper/E3RTUKRB

@misc{pith2026250909446,
  author       = {Pith},
  title        = {Pith review of: $p$-adic Higher Green's Functions for Stark-Heegner Cycles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E3RTUKRB}},
  note         = {Machine review of arXiv:2509.09446}
}

abstract

Heegner cycles are higher weight analogues of Heegner points. Their arithmetic intersection numbers also appear as Fourier coefficients of modular forms and often belong to abelian extensions of imaginary-quadratic fields. Rotger and Seveso propose a precise recipe for the $p$-adic Abel-Jacobi images of cycle classes whose existence is predicted by conjectures of Bloch and Beilinson and which would be a real-quadratic analogue to Heegner cycles: the Stark-Heegner cycles of the title. In this paper, we generalize Darmon-Vonk's theory of rigid meromorphic cocycles to higher weight, producing a higher Green's pairing of real-quadratic divisors on the $p$-adic upper half-plane, which seems to be the real-quadratic analogue of the pairing of Heegner cycles. Computation of these values for "principal cycles'' gives evidence that they lie in abelian extensions of real-quadratic fields. The algebraicity of certain values of the higher Green's function is indirect evidence for the existence of algebraic Stark-Heegner cycles.

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [1]

Modular forms in characteristic l and special values of their L -functions

Avner Ash and Glenn Stevens. Modular forms in characteristic l and special values of their L -functions. Duke Math. J. , 53(3):849--868, 1986

1986

[2] [2]

The rationality of S tark- H eegner points over genus fields of real quadratic fields

Massimo Bertolini and Henri Darmon. The rationality of S tark- H eegner points over genus fields of real quadratic fields. Ann. of Math. (2) , 170(1):343--370, 2009

2009

[3] [3]

C M values of higher automorphic G reen functions for orthogonal groups

Jan Hendrik Bruinier, Stephan Ehlen, and Tonghai Yang. C M values of higher automorphic G reen functions for orthogonal groups. Invent. Math. , 225(3):693--785, 2021

2021

[4] [4]

Bruinier, Yingkun Li, and Tonghai Yang

Jan H. Bruinier, Yingkun Li, and Tonghai Yang. Deformations of theta integrals and a conjecture of G ross- Z agier, 2022

2022

[5] [5]

Invariant L et s\'erie sp\'eciale p -adique

Christophe Breuil. Invariant L et s\'erie sp\'eciale p -adique. Ann. Sci. \'Ecole Norm. Sup. (4) , 37(4):559--610, 2004

2004

[6] [6]

S\'erie sp\'eciale p -adique et cohomologie \'etale compl\'et\'ee

Christophe Breuil. S\'erie sp\'eciale p -adique et cohomologie \'etale compl\'et\'ee. Ast\'erisque , (331):65--115, 2010

2010

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Robert F. Coleman. Dilogarithms, regulators and p -adic L -functions. Invent. Math. , 69(2):171--208, 1982

1982

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Choie and D

Yj. Choie and D. Zagier. Rational period functions for PSL (2, Z) . In A tribute to E mil G rosswald: number theory and related analysis , volume 143 of Contemp. Math. , pages 89--108. Amer. Math. Soc., Providence, RI, 1993

1993