{"id":"f9392484-973d-4d53-8302-1a7ea96af65a","arxiv_id":"2412.14289","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The mod-5 representation on H^3 of the Calabi-Yau threefold Y79 is shown to be isomorphic to the mod-5 residual representation of the paramodular form F79.","lead":"A team proved that the mod-5 Galois representation on the third cohomology of a specific Calabi-Yau threefold is isomorphic to the one attached to a genus-2 Siegel modular form. This is a step toward a conjecture of Golyshev and van Straten, connecting geometry to modular forms.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The proof of Theorem 1.2 rests on an unprinted finite check of 1313 prime ideals whose completeness and correctness are not reproducible from the paper; if any prime is missing or an eigenvalue is miscomputed, Corollary 1.3 can fail.","rationale":"I read the full manuscript and the central claim carefully. The main theorem chain is: Theorem 1.2 proves a mod-λ congruence between Hilbert newforms f79 and h79; Theorem 1.1 proves a mod-λ congruence between the paramodular form F79 and the Johnson-Leung-Roberts lift of h79; together with the Golyshev-van Straten description of H^3_et(Y79, F5), these imply Corollary 1.3. The geometric and algebraic steps around the computational checks are largely standard and internally consistent: the intersection argument in Lemma 2.1 is sound because the one-dimensional intersection is a simultaneous eigenspace; the Sturm-bound argument in Section 5 is supported by cited results on Hilbert modular surfaces; and the use of partial Hasse invariants and theta operators is documented. I looked for an internal inconsistency, such as in the inertness claim for 79 in Q(ζ5)/Q(√5) or in the irreducibility assumptions, and found no clear mathematical error. The reader's weakest-assumption identification is accurate: Lemma 5.2 is the point where the proof reduces to a large finite computation, and the paper does not make that computation auditable in the text. The statement that the congruence is 'visibly satisfied' is not a substitute for a reproducible verification. Because the omitted data are central rather than auxiliary, the CONDITIONAL verdict is appropriate; I would not move it to ACCEPT without independent verification of the 1313-prime check, nor to REJECT without evidence of an actual failure.","tokens_in":18432,"tokens_out":9313,"duration_ms":89131,"concrete_test":"Recompute the finite check in Lemma 5.2 independently: (1) generate all totally positive ξ in the inverse different with tr(ξ) < 97, factor √5ξ, and form the list of prime ideals; verify it has 1313 entries and matches the output at [14]; (2) for each p on the list, compute μ_p(f79) and μ_p(h79) modulo λ using an independent implementation (e.g., Sage/PARI or LMFDB data) and verify the congruence; (3) as a stronger check, compare the normalized q-expansion coefficients aξ and bξ mod λ directly for all tr(ξ) < 97, rather than only prime ideals, to rule out a normalization or multiplicativity error in passing from coefficients to Hecke eigenvalues.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing step is Lemma 5.2 in the proof of Theorem 1.2: after the Sturm-bound reduction, the equality of the relevant q-expansions is claimed to follow from the congruence μ_p(f79) ≡ μ_p(h79) (mod λ) for all prime ideals p ≠ (√5) dividing elements √5ξ with ξ ∈ d^{-1}, ξ ≫ 0, tr(ξ) < 97. The paper reports that Magma collected 1313 such p and that the congruence is 'visibly satisfied' in the output, but neither the list of primes nor the code is reproduced in the manuscript; only a URL is given, without a commit hash or checksum. This creates two distinct failure modes. First, if the script that builds the list omits any prime dividing one of these elements, the Sturm criterion is not met and a nonzero q-expansion could survive. Second, if any of the 1313 eigenvalue pairs in F(h79) reduced modulo λ is computed incorrectly, the displayed congruence could be spurious. Since Theorems 1.1 and 1.2 combine to give Corollary 1.3, both failure modes would invalidate the central claim. No internal inconsistency in the surrounding geometric argument was found; the omitted proof of Lemma 4.1 is a lesser concern because it is a standard q-expansion principle with a cited analogue.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":18690,"tokens_out":6523,"duration_ms":57696,"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":[{"comment":"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.","section":"Section 5, Lemma 5.2"},{"comment":"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.","section":"Section 2.4, Lemma 2.1"},{"comment":"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.","section":"Section 5, proof of Lemma 5.2"}],"minor_comments":[{"comment":"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.","section":"Section 4, Lemma 4.1"},{"comment":"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.","section":"Section 5, after equation (3)"},{"comment":"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.","section":"Section 6, Theorem 6.1"},{"comment":"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.","section":"Section 1, Remark 1.4"}],"recommendation":"major_revision","confidential_remarks":"The external computations appear central to both main theorems. I would ask the editor to require the authors to deposit the code and outputs in a stable repository with checksums and package versions. The self-citation [13] is used as a theorem with its own proof, so there is no circularity. If the computational gaps are closed and the cusp argument is clarified, the paper would be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The thing to know: this paper proves a weak form of the Golyshev–van Straten conjecture for Y79, mod 5. That is a real result. It shows the mod-5 Galois representation on H^3 of Y79 is isomorphic to the residual representation attached to the non-lift paramodular form F79. The proof is a chain of new congruences: between Hilbert forms f79 and h79 over Q(√5), and between F79 and the Johnson–Leung–Roberts lift of h79. Nothing here is fitted; the forms are fixed by prior literature and the congruences are verified, not chosen.\n\nWhat the paper does well: it extends the orthogonal-modular-forms method to non-squarefree level 79×5^2, and it handles the ramified characteristic 5 case using generalized partial Hasse invariants and partial theta operators from Diamond and Reduzzi–Xiao. The structural argument is coherent: Brauer–Nesbitt plus trace checks, the intersection argument for V1∩V2, and the Sturm bound reduction to a finite list. The authors also include a secondary mod 2 congruence, which is a nice bonus. The paper is honest about what is checked by machine and what is proved by hand.\n\nThe soft spot is exactly where the stress-test note lands. Lemma 5.2, the load-bearing step for Theorem 1.2, reduces to a Magma computation over 1313 prime ideals, and the paper does not list those primes or the code inline. The authors point to an external URL with code and output, but there is no commit hash or checksum, so a referee cannot independently reproduce the check from the manuscript alone. If any prime is missing or an eigenvalue is miscomputed, the Sturm argument does not close and Corollary 1.3 could fail. I think this is a genuine reproducibility gap, but it is not evidence of a mathematical error. The surrounding arguments are careful, and the omitted proof of Lemma 4.1 is minor—it is a standard q-expansion principle with a cited analogue.\n\nWho this is for: arithmetic geometers and computational number theorists working on paramodular forms and Calabi–Yau modularity. It deserves a serious referee. The right outcome is probably acceptance after the authors provide the full list of primes, the code, and a reproducible packaging of the computation—ideally with a checksum or a way to rerun it without hidden state.\n\nRecommendation: send it to review, and ask the referees to verify the computational step or at least confirm the script's completeness.\n\nBest,\n\n[You]","headline":"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.","tokens_in":19258,"tokens_out":1879,"would_cite":true,"duration_ms":15605,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["11F33","11F41","11F46","11G40","14J32"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Hilbert modular form","paramodular form","Calabi-Yau threefold","Galois representation","Hecke eigenvalue congruence","Golyshev-van Straten conjecture","Siegel modular form","modularity"],"falsifier":"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.","tokens_in":18217,"feed_emoji":"🧮","tokens_out":13993,"duration_ms":106275,"temperature":0.7,"pith_summary":"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.","feed_headline":"Cohomology of Calabi-Yau Y79 matches a Siegel modular form mod 5","feed_subtitle":"Two mod-5 congruences link the threefold's cohomology to a Siegel form, a step toward Golyshev–van Straten.","key_machinery":"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 $\\lambda$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Constructs the Calabi-Yau family and proves that $H^3(Y_{79},\\mathbb{F}_5)$ is induced from an elliptic-curve 5-torsion representation; this is the geometric input for Corollary 1.3 and the source of the conjecture.","marker":"[18]"},{"why":"Shows elliptic curves over real quadratic fields are modular, so the elliptic curve attached to $f_{79}$ corresponds to a Hilbert newform.","marker":"[16]"},{"why":"Defines the Johnson-Leung-Roberts lift, whose Galois representation is the induction of $\\rho_{h_{79},\\lambda}$ and which bridges the Hilbert and Siegel sides.","marker":"[22]"},{"why":"Develops the orthogonal-modular-form realization of paramodular forms at level $79\\cdot 5^2$ used in the proof of Theorem 1.1.","marker":"[13]"},{"why":"Supplies the Sturm bound on Hilbert modular surfaces and the intersection-theoretic method at the core of the proof of Theorem 1.2.","marker":"[6]"},{"why":"Introduces the geometric partial theta operator and weight-shifting in characteristic $p$, allowing comparison of forms of different weights.","marker":"[9]"},{"why":"Provides the partial Hasse invariants and the triviality facts for line bundles needed to bring the two forms to a common parallel weight.","marker":"[11]"},{"why":"Gives partial Hasse invariants on splitting models of Hilbert modular varieties, needed because 5 is ramified in $\\mathbb{Q}(\\sqrt{5})$.","marker":"[33]"},{"why":"The computer algebra system used to compute Hecke eigenvalues and to run the 1313-prime verification that closes the Sturm-bound argument.","marker":"[4]"},{"why":"Provides the explicit non-lift paramodular eigenform $F_{79}$ in $S_3(K(79))$ and its mod 2 reducibility data.","marker":"[27]"}],"fun_headline_variants":["Mod-5 congruence links Calabi-Yau threefold to Siegel form","Residual mod-5 match for Calabi-Yau cohomology","Calabi-Yau cohomology mod 5 equals Siegel residual","H^3 mod 5 of Y79 matches Siegel form","Weak Golyshev–van Straten via mod-5 congruence"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Mod-5 congruence links Calabi-Yau threefold to Siegel form","Residual mod-5 match for Calabi-Yau cohomology","Calabi-Yau cohomology mod 5 equals Siegel residual","H^3 mod 5 of Y79 matches Siegel form","Weak Golyshev–van Straten via mod-5 congruence"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00063,"raw_usage":{"total_tokens":3011,"prompt_tokens":1149,"completion_tokens":1862,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":765,"completion_tokens_details":{"reasoning_tokens":1770}},"tokens_in":765,"tokens_out":1862,"duration_ms":13208,"temperature":1.0,"reasoning_tokens":1770,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T12:22:12.084916+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Golyshev, D","cited_arxiv_id":null,"evidence_quote":"Constructs the Calabi-Yau family and proves that $H^3(Y_{79},\\mathbb{F}_5)$ is induced from an elliptic-curve 5-torsion representation; this is the geometric input for Corollary 1.3 and the source of the conjecture."},{"cited_title":"Freitas, B","cited_arxiv_id":null,"evidence_quote":"Shows elliptic curves over real quadratic fields are modular, so the elliptic curve attached to $f_{79}$ corresponds to a Hilbert newform."},{"cited_title":"Johnson-Leung, B","cited_arxiv_id":null,"evidence_quote":"Defines the Johnson-Leung-Roberts lift, whose Galois representation is the induction of $\\rho_{h_{79},\\lambda}$ and which bridges the Hilbert and Siegel sides."},{"cited_title":"Dummigan, A","cited_arxiv_id":null,"evidence_quote":"Develops the orthogonal-modular-form realization of paramodular forms at level $79\\cdot 5^2$ used in the proof of Theorem 1.1."},{"cited_title":"Burgos Gil, A","cited_arxiv_id":null,"evidence_quote":"Supplies the Sturm bound on Hilbert modular surfaces and the intersection-theoretic method at the core of the proof of Theorem 1.2."},{"cited_title":"Diamond, Geometric weight-shifting operators on Hil bert modular forms in characteristic p, J","cited_arxiv_id":null,"evidence_quote":"Introduces the geometric partial theta operator and weight-shifting in characteristic $p$, allowing comparison of forms of different weights."},{"cited_title":"Diamond, S","cited_arxiv_id":null,"evidence_quote":"Provides the partial Hasse invariants and the triviality facts for line bundles needed to bring the two forms to a common parallel weight."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives partial Hasse invariants on splitting models of Hilbert modular varieties, needed because 5 is ramified in $\\mathbb{Q}(\\sqrt{5})$."},{"cited_title":"Bosma, J","cited_arxiv_id":null,"evidence_quote":"The computer algebra system used to compute Hecke eigenvalues and to run the 1313-prime verification that closes the Sturm-bound argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the explicit non-lift paramodular eigenform $F_{79}$ in $S_3(K(79))$ and its mod 2 reducibility data."}],"review_version":1}