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Residual paramodularity of a certain Calabi-Yau threefold

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Pith's one-line read The paper proves that the mod-5 Galois representation on the third cohomology of the Calabi-Yau threefold $Y_{79}$ is isomorphic to the mod-5 residual representation of a genus-2 Siegel modular form $F_{79}$, a weak form of the…

desk verdict A genuinely new residual modularity result for a single Calabi-Yau threefold, built on a coherent geometric argument but resting on a load-bearing finite computation that the paper does not fully reproduce. read the letter →

arxiv 2412.14289 v1 pith:BW3KNQAO submitted 2024-12-18 math.NT

Neil Dummigan , Gonzalo Tornaría This is my paper · ORCID
classification math.NT MSC 11F3311F4111F4611G4014J32
keywords HilbertmodularformparamodularCalabi-YauthreefoldGaloisrepresentationHeckeeigenvaluecongruenceGolyshev-vanStratenconjectureSiegelmodularity
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

The paper proves that the mod-5 Galois representation on the third cohomology of a particular Calabi-Yau threefold, $Y_{79}$, is isomorphic to the mod-5 residual representation attached to a genus-2 Siegel modular form $F_{79}$ of paramodular level 79. The route is a chain of Hecke-eigenvalue congruences: a weight-$(2,2)$ Hilbert newform $f_{79}$ is congruent modulo a divisor $\lambda$ of 5 to a weight-$(2,4)$ newform $h_{79}$ over $\mathbb{Q}(\sqrt{5})$, and $F_{79}$ is congruent modulo $\lambda$ to the Johnson-Leung-Roberts lift of $h_{79}$. Because the cohomology representation is induced from the elliptic-curve representation attached to $f_{79}$, these congruences force the isomorphism $H^3_{\acute{e}t}(Y_{79}\otimes\bar{\mathbb{Q}}, \mathbb{F}_5)\cong\rho_{F_{79},5}$. This is a weak form of the Golyshev-van Straten conjecture that $Y_{79}$ is paramodular. The proof combines a Sturm-bound argument on a Hilbert modular surface with a finite computer verification over 1313 prime ideals.

What carries the argument

The argument is carried by two linked mechanisms. The first is a chain of Hecke-eigenvalue congruences: for the unique divisor $\lambda$ of 5 in the coefficient field $F(h_{79})$, the eigenvalues of $f_{79}$ and $h_{79}$ agree in $\mathbb{F}_5$ away from the ramified prime, and the eigenvalues of $F_{79}$ agree with those of the Johnson-Leung-Roberts lift $\mathrm{JR}(h_{79})$, a map that turns a Hilbert newform into a genus-2 Siegel paramodular form with the same spin $L$-function. The second is a geometric Sturm-bound proof in characteristic 5: a Sturm bound is a limit on how many Fourier coefficients must agree before two modular forms are identical. Because 5 is ramified in $\mathbb{Q}(\sqrt{5})$, the authors work on Pappas-Rapoport Hilbert modular surfaces and use generalised partial Hasse invariants $H_1,H_2$ and a partial $\theta$ operator $\Theta$ to shift weights and kill unwanted Fourier coefficients, reducing the desired equality to the vanishing of one form $G$ of weight $(8J,8J)$. Intersection theory on the compactified Hilbert modular surface gives a bound $C\le 96$ on the order of vanishing at cusps, while the required vanishing needs $C\ge 97$; the gap is closed by checking, for all 1313 prime ideals dividing $\sqrt{5}\xi$ with $\mathrm{tr}(\xi)<97$, that the corresponding Hecke eigenvalues are congruent modulo $\lambda$.

What would settle it

Run an independent, fully documented computation of $\mu_p(f_{79})$ and $\mu_p(h_{79})$ modulo $\lambda$ for every prime ideal $p$ dividing $\sqrt{5}\xi$ with $\xi\gg0$ and $\mathrm{tr}(\xi)<97$, and verify both that the list of such primes is complete and that each congruence holds; a single discrepancy, or a missing prime ideal, would break the $C\ge 97$ step and hence Theorem 1.2.

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Extended reading notes

Core claim

The central assertion is Corollary 1.3: the 4-dimensional representation of $\mathrm{Gal}(\bar{\mathbb{Q}}/\mathbb{Q})$ on $H^3_{\acute{e}t}(Y_{79}\otimes\bar{\mathbb{Q}}, \mathbb{F}_5)$ is isomorphic to the mod-5 residual representation $\rho_{F_{79},5}$ attached to the non-lift Hecke eigenform $F_{79}\in S_3(K(79))$, where $K(79)$ is the paramodular subgroup of $\mathrm{Sp}_4(\mathbb{Q})$. The authors do not prove full 5-adic modularity; they prove residual modularity modulo 5. The deduction runs through two theorems. Theorem 1.2 establishes a mod-$\lambda$ congruence of Hecke eigenvalues between the Hilbert newforms $f_{79}$ (weight $(2,2)$) and $h_{79}$ (weight $(2,4)$) over $\mathbb{Q}(\sqrt{5})$, and Theorem 1.1 establishes a mod-$\lambda$ congruence between $F_{79}$ and the Johnson-Leung-Roberts lift $\mathrm{JR}(h_{79})$ at paramodular level $79\cdot 5^2$. Together with the Golyshev-van Straten description of $H^3_{\acute{e}t}(Y_{79},\mathbb{F}_5)$ as the induction of the 5-torsion representation of an elliptic curve over $\mathbb{Q}(\sqrt{5})$, the congruences imply the corollary. The paper notes that the conclusion is residual in a strong sense: $F_{79}$ is not a lift of a Hilbert modular form, only congruent modulo 5 to one, and existing modularity lifting theorems do not yet upgrade the residual isomorphism to the 5-adic representation.

Load-bearing premise

The proof of Theorem 1.2 rests on a finite computer calculation: the authors state that the congruence of Hecke eigenvalues is visibly satisfied for all 1313 prime ideals on their list and supply the code and output only at a web address, so an error or omission in that computation would invalidate Lemma 5.2 and, with it, Corollary 1.3.

Editorial extensions

If this is right

  • The mod-5 cohomology representation of $Y_{79}$ is realized by the paramodular eigenform $F_{79}$, so $Y_{79}$ is residually paramodular modulo 5.
  • The weight-$(2,2)$ and weight-$(2,4)$ Hilbert newforms $f_{79}$ and $h_{79}$ have isomorphic residual Galois representations over $\mathbb{Q}(\sqrt{5})$, giving congruent traces of Frobenius at every unramified prime away from the ramified prime.
  • The non-lift form $F_{79}$ is congruent modulo 5 to a Johnson-Leung-Roberts lift, even though it is not itself such a lift, so the residual data do not come from a simpler Hilbert-modular source.
  • The same Sturm-bound framework, adapted to characteristic 2, yields a mod-$q_1$ congruence between $f_{79}$ and $h_{79}$ for all primes away from 2, a secondary residue congruence.
  • The residual isomorphism is a first step toward full 5-adic modularity of $Y_{79}$; the paper observes that currently available modularity lifting theorems do not suffice to take that step.

Reading between the lines

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

  • Extension: The authors do not claim the 5-adic version; a natural next step would be a modularity-lifting theorem for residually reducible inductions, or a test of a mod-$5^n$ refinement against the Euler factors of $Y_{79}$ and $F_{79}$.
  • Extension: The same Sturm-bound setup may transfer to other fibres $Y_t$ of the Golyshev-van Straten pencil; the paper's remarks about level 431 and the negative check at level 31 suggest the congruence phenomenon is tied to the geometry of the Apery family rather than an accident of $t=-1$.
  • Extension: Because the ramified prime 5 is built into the construction through the Apery family's 5-torsion, extending the result to other primes would likely require a new source of torsion rather than a routine recomputation.
  • Extension: The mod-2 statement is a by-product of the same machinery; an independent test would be to check whether the associated elliptic curves over $\mathbb{Q}(\sqrt{5})$ indeed have fully rational 2-torsion, as the paper's explanation predicts.
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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

3 major / 4 minor

Summary. The paper proves congruences of Hecke eigenvalues between Hilbert newforms f79 and h79 over Q(√5), modulo a degree-one prime λ above 5 and modulo a prime q1 above 2. The mod-5 congruence is combined with a congruence between the paramodular form F79 and the Johnson-Leung-Roberts lift of h79 to prove that the mod-5 Galois representation on H^3_et(Y79 ⊗ Q̄, F5) is isomorphic to the residual representation ρ_{F79,5}. This establishes a weak form of the Golyshev–van Straten conjecture for Y79. The proof uses a Sturm-bound argument on a Hilbert modular surface in characteristic 5, with generalized partial Hasse invariants and partial theta operators, reducing the equality of q-expansions to a finite check of 1313 prime ideals.

Significance. If the finite computations are correct, Corollary 1.3 is a notable result: Y79 becomes the first Calabi-Yau threefold for which residual paramodularity is proved, and the method demonstrates how to handle the ramified case ℓ=5 via Pappas–Rapoport models and partial theta operators. The paper is careful to reduce the proof to finite computations and makes code and outputs available, which is commendable. However, the external data are not archived with checksums, so the reproducibility of the key checks is currently incomplete.

major comments (3)
  1. [Section 5, Lemma 5.2] The proof of Theorem 1.2 depends on a finite computation that is not reproducible from the manuscript. The paper states that Magma collected 1313 prime ideals and that 'looking at the output, the congruence is visibly satisfied,' but neither the list of primes nor the verification code is included; only a URL ([14]) is given, without a commit hash or checksum. This is load-bearing: a missing prime or a miscomputed eigenvalue in the list would invalidate the Sturm-bound conclusion and hence Corollary 1.3. Please provide the code and full output as supplementary material, or at least a machine-readable list of the 1313 primes with the computed eigenvalues.
  2. [Section 2.4, Lemma 2.1] The proof of Theorem 1.1 relies on the computed dimensions of V, V1, V2, and V1⊗F5∩V2⊗F5, obtained with G. Rama's packages. The code and output are again only available at [14] with no version or commit information. Because Theorem 1.1 is an essential input to Corollary 1.3, please include the code (or a stable archive with checksums and package versions) so that the computation can be independently rerun.
  3. [Section 5, proof of Lemma 5.2] The reduction from all ten cusps to the single cusp D1 is not justified. The text says that because the auxiliary level at 3 creates ten cusps instead of one, it suffices to prove div(G) ≥ 97D1. But G, while invariant under U', is not obviously symmetric under the permutations of the ten cusps; the product over the cosets {gi} for the level-n part does not, as written, imply that the order at every other cusp is at least the order at D1. Please supply a proof or a precise reference for this implication; otherwise the Sturm bound may only control one cusp.
minor comments (4)
  1. [Section 4, Lemma 4.1] The q-expansion principle is stated without proof. Since the manuscript is otherwise careful to reduce to finite checks, a proof or a more precise citation (with a statement of why the cited theorem applies in the ramified, Pappas–Rapoport setting) would improve completeness.
  2. [Section 5, after equation (3)] The notation 'div(G) ≥ 97D' is not defined; since D is a sum of divisors, please clarify that this means the order of G at each irreducible component of D is at least 97.
  3. [Section 6, Theorem 6.1] The proof is only sketched; a sentence pointing to the exact analogous steps in the proof of Theorem 1.2 (with the modifications for two theta operators) would help the reader verify the bound C ≤ 144.
  4. [Section 1, Remark 1.4] The sentence 'So this does not rule out...' appears to be a non sequitur; the preceding observation about f31 and h31 is about a different level, so it is unclear how it relates to the analogues for all nonzero rational t. Please expand.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the mod-5 congruences are proved by Sturm-bound finite verification and an independent prior correspondence; no fitted parameter or input-dependent prediction is disguised as a result.

full rationale

The paper's central claim (Corollary 1.3) is assembled from two congruences. Theorem 1.2 is proved by reducing an equality of q-expansions in characteristic 5 to a Sturm-type bound, then verifying the required Hecke-eigenvalue congruences for the 1313 prime ideals listed by Magma (Lemma 5.2). These eigenvalues are intrinsic data attached to the fixed forms f79 and h79, not parameters fitted to the desired conclusion; the verification is a finite check, and the paper does not define the forms in terms of the congruence. Theorem 1.1 is proved by realizing both F79 and JR(h79) in a space of orthogonal modular forms via the cited correspondence [13, Theorem 9.9, Theorem 10.1], where the identification of the relevant subspaces uses independently computed T2 eigenvalues and dimensions. Although [13] involves the same authors, it is a previously proven theorem with its own proofs and code, and its assumptions do not contain the present congruence; hence it is independent evidence rather than circular self-citation. The remaining ingredients (the Golyshev--van Straten theorem [18], the Freitas--Le Hung--Siksek modularity theorem, the Johnson-Leung--Roberts lift, and the identification of F79 from [27]) are external results. No step reduces, by construction or by definition, to the claim being proved, and no fitted parameter is renamed as a prediction. The finite Magma computation is not reproducible from the printed paper alone, but that is a correctness/expository risk, not circularity.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The proof imports substantial machinery (Hilbert modular forms, paramodular forms, orthogonal modular forms, and characteristic-p geometric models) and large computations. No new free parameters or invented entities are introduced. The main risk is the correctness of the external computation outputs and the cited structural theorems.

assumptions (6)
  • domain assumption Magma and PARI/Sage outputs used for dimensions, eigenvalues, and congruence checks are correct.
    Used in Section 2.1, Section 2.4, Section 5, and Section 6; the paper provides external data at [14] but no formal proof of the computations.
  • domain assumption The correspondence (1) between weight-3 paramodular forms of level 79×5^2 and orthogonal modular forms from [13] holds exactly as stated.
    Invoked in Section 2.4; the argument identifying V1 and V2 depends on it.
  • domain assumption The generalized partial Hasse invariants and partial theta operators exist with the stated weights and q-expansion effects for ramified characteristic ℓ=5.
    Used in Section 4 to transform f79 and h79 into a common weight; built on [9], [11], [33], and the lecture [10].
  • domain assumption The intersection-theoretic Sturm bound of Burgos Gil and Pacetti carries over to the Pappas-Rapoport model in characteristic 5, yielding C ≤ 96.
    Used in Lemma 5.1; the paper adapts [6, Appendix A and B] and [23].
  • domain assumption The modularity theorem for elliptic curves over real quadratic fields [16] applies to EL,t/u.
    Used in Section 1 to identify f79 with the associated Hilbert newform.
  • domain assumption The dimension formulas of Ibukiyama and the uniqueness of F79 from [27] identify the non-lift eigenform.
    Used in Section 1 to set up the conjecture and in Section 2.3.

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Pith. "Pith review of Residual paramodularity of a certain Calabi-Yau threefold." pith.science (2026). https://pith.science/paper/BW3KNQAO

@misc{pith2026241214289,
  author       = {Pith},
  title        = {Pith review of: Residual paramodularity of a certain Calabi-Yau threefold},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BW3KNQAO}},
  note         = {Machine review of arXiv:2412.14289}
}
abstract

We prove congruences of Hecke eigenvalues between cuspidal Hilbert newforms $f_{79}$ and $h_{79}$ over $F=\mathbb Q(\sqrt{5})$, of weights (2,2) and (2,4) respectively, level of norm 79. In the main example, the modulus is a divisor of 5 in some coefficient field, in the secondary example a divisor of 2. The former allows us to prove that the 4-dimensional mod-5 representation of $\mathrm{Gal}(\overline{\mathbb Q} / \mathbb Q)$ on the 3rd cohomology of a certain Calabi-Yau threefold comes from a Siegel modular form $F_{79}$ of genus 2, weight 3 and paramodular level 79. This is a weak form of a conjecture of Golyshev and van Straten. In aid of this, we prove also a congruence of Hecke eigenvalues between $F_{79}$ and the Johnson-Leung-Roberts lift $\mathrm{JR}(h_{79})$, which has weight 3 and paramodular level $79\times 5^2$.

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