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On Some Hypergeometric Modularity Conjectures of Dawsey and McCarthy

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Pith's one-line read The paper proves five Dawsey–McCarthy hypergeometric modularity conjectures by constructing explicit weight-three Hecke eigenforms from cubic theta eta-quotients.

desk verdict Solid K3 construction and applications, but the proof of the (3,3) and (6,3) cases misapplies Theorem 2 because the individual K3 eta-quotients are not Hecke eigenforms. read the letter →

arxiv 2507.19971 v1 pith:KXJVQZPK submitted 2025-07-26 math.NT

Brian Grove This is my paper · ORCID
classification math.NT MSC 05C9011F1111R3233C0533C60
keywords hypergeometricmodularityeta-quotientsHeckeeigenformscubicthetafunctionsgeneralizedPaleygraphsspecialL-valuesDawsey–McCarthyconjecturesExplicitMethod
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 proves five of the hypergeometric modularity conjectures proposed by Dawsey and McCarthy. For each pair $(u,v)$ in $\{(2,3),(3,3),(4,3),(6,3),(12,3)\}$, it establishes that the finite-field hypergeometric sum $H_p(HD_{DM}(u,v);1;\mathfrak{p})$ equals a character twist $\psi_{(u,v)}(\mathfrak{p})$ of the $p$-th Fourier coefficient of an explicit weight-three Hecke eigenform, at every prime ideal $\mathfrak{p}$ above a prime $p \equiv 1 \pmod M$. The engine is a new family of eta-quotients, the $K_3$ functions, built from the weight-one cubic $\theta$ functions and a cubic analogue of Jacobi's identity; in four Galois families these quotient functions are completed to the target eigenforms by explicit Hecke operator computations. The identities are then used to derive special $L$-value formulas for the new eigenforms and an exact formula for the number of order-four complete subgraphs in cubic generalized Paley graphs. This matters because hypergeometric sums are hard to evaluate directly, and matching them to modular form coefficients creates an explicit dictionary between finite-field character sums and modular arithmetic.

What carries the argument

The central object is the $K_3$ family of eta-quotients $K_3(r,1)(\tau)=\eta(\tau)^{9-12r}\eta(3\tau)^{12r-3}$ for $r\in\{j/12:1\le j\le 11\}$, built from the weight-one cubic $\theta$ functions $a,b,c$ satisfying $a^3=b^3+c^3$ and the $\Gamma_0(3)$ Hauptmodul $t_3=(c/a)^3$. After the rescaling $\tau\mapsto N_{K_3}(r)\tau$ these are weight-three holomorphic cusp forms, and in the four Galois families of Table 1 they are combined, with constants determined by explicit Hecke operator computations, into the target Hecke eigenforms $f^\sharp$, meaning modular forms that are simultaneous eigenvectors for the Hecke operators. The proof mechanism is the two-part Explicit Hypergeometric Modularity Method: the Euler-integral and Schwarz-map computation produces the modular form from the hypergeometric datum, and the chain of $p^2$ supercongruences identifies $H_p(HD;1;\mathfrak{p})$ with $a_p(f^\sharp)$ up to the character twist.

What would settle it

Compute the two sides of any one of the five identities at a prime $p$ with $13\le p\le 29$ and $p\equiv 1\pmod M$; a single mismatch falsifies the all-primes claim. Separately, a direct computation of the asserted Hecke relations, for example $T_5^2=45$, $T_7^2=-135$, $T_{11}^2=-243$, and $T_5T_7=5T_{11}$, on the relevant $K_3$ subspace would confirm or refute the eigenform identification itself.

Watch

Extended reading notes

Core claim

Theorem 1 asserts that for the five pairs $(u,v)$ listed above, $H_p(HD_{DM}(u,v);1;\mathfrak{p}) = \psi_{(u,v)}(\mathfrak{p})\, a_p(f^\sharp_{HD_{DM}(u,v)})$ at every prime ideal $\mathfrak{p}$ above each prime $p\equiv 1 \pmod M$, where $f^\sharp_{HD_{DM}(u,v)}$ are the explicit weight-three Hecke eigenforms $f_{12.3.c.a}$, $f_{27.3.b.b}$, $f_{16.3.c.a}$, $f_{108.3.c.b}$, and $f_{432.3.g.e}$. The paper constructs the $K_3$ eta-quotient family $K_3(r,1)(\tau)=\eta(\tau)^{9-12r}\eta(3\tau)^{12r-3}$ for $r=j/12$ with $1\le j\le 11$, proves by the eta-quotient criterion that these are weight-three holomorphic cusp forms after scaling $\tau\mapsto N_{K_3}(r)\tau$, and identifies the eigenform completions in the four Galois families of Table 1 using explicit Hecke operator constants. On the hypergeometric side, the Euler integral formula and the Schwarz map connect the same data to the Hauptmodul $t_3(\tau)=27\eta(3\tau)^9/(3\eta(3\tau)^3+\eta(\tau/3)^3)^3$, so the two-part Explicit Hypergeometric Modularity Method congruence argument closes the loop. As applications, Lemma 3 and Corollary 1 express the special values $L(f,1)$ as explicit combinations of hypergeometric periods, and Corollary 2 converts the $(3,3)$ identity into a formula for $K_4(G_3(p))$, the number of order-four cliques in the cubic generalized Paley graph.

Load-bearing premise

The load-bearing premise is that the stated Hecke operator actions identifying the five eigenforms are correct, together with the unstated assumption that the general method, proved only for primes above $29$, still holds at the small primes claimed in Theorem 1.

Editorial extensions

If this is right

  • The five identities supply explicit replacements for the corresponding conjectured formulas: each $H_p(HD_{DM}(u,v);1;\mathfrak{p})$ can be computed from the Fourier coefficients of a known weight-three eigenform at primes $p\equiv 1\pmod M$.
  • Lemma 3 gives $3P_2(HD_{K_3}(r,1);1)=2\cdot 3^{3r-1/2}N\,\pi\, L(K_3(r,1)(N\tau),1)$, so the special $L$-values of the five eigenforms are finite combinations of hypergeometric periods as listed in Corollary 1.
  • The Kummer transformation (6.9) yields the companion eta-quotient family $K_3^{kmr}(r)=\eta(\tau)^{1-12r}\eta(3\tau)^{12r+5}$, which coincides with $K_3$ values at $r=1/12,1/6,1/4$ but is generally not holomorphic.
  • For primes $p\equiv 1\pmod 6$, the $(3,3)$ identity gives an exact formula for $K_4(G_3(p))$ in terms of $a_p(f_{27.2.a.a})$ and $a_p(f_{27.3.b.b})$, confirming the Dawsey–McCarthy graph conjecture.
  • The Atkin–Lehner involution $\tau\mapsto -1/(3\tau)$ maps $K_3(r,1)(\tau)$ to a constant multiple of $K_3(1-r,1)(\tau)$, so the $K_3$ family is closed under this symmetry.

Reading between the lines

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

  • Going beyond the paper, the same $K_3$ machinery may apply to other hypergeometric data from the original survey, such as the remaining $v\neq 3$ pairs, whenever the datum admits a Hauptmodul substitution; a direct test would be to run the construction on those pairs and compare the resulting sums with known eigenform coefficients.
  • The paper's general theorem is stated for primes $p>29$, so a reader who wants to use Theorem 1 at small primes should verify the five identities numerically there; the paper does not report those checks.
  • The $L$-value relations in Corollary 1, combined with the Kummer transformation (6.9), suggest further identities among the hypergeometric periods $3P_2(1)$ and the special $L$-values of the $K_3$ eigenforms that are not spelled out in the paper and could be tested at the listed $r$-values.
  • Because the $(3,3)$ identity feeds directly into the graph count $K_4(G_3(p))$, the same hypergeometric-to-modular dictionary may give modular-form formulas for larger cliques in generalized Paley graphs, though the paper only treats order four.
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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 / 3 minor

Summary. The paper introduces two families of weight-three eta-quotients, the K3 and K^{kmr}_3 functions, derived from Borwein's cubic theta functions, and uses them with the Explicit Hypergeometric Modularity Method (EHMM) from the author's prior work to prove five Dawsey–McCarthy hypergeometric modularity conjectures (Theorem 1). It also derives special L-value identities (Lemma 3, Corollary 1), Kummer and Atkin–Lehner transformation laws, and a corollary expressing K4(G3(p)) in terms of modular form coefficients (Corollary 2).

Significance. The K3 construction is a clean and explicit cubic analog of the K2 construction, and the L-value and Paley-graph applications are natural. The paper is transparent about relying on the prior EHMM [3,4], and the target eigenforms are given by explicit LMFDB labels with no fitted parameters. If Theorem 1 is established, it would provide a substantial new family of explicit hypergeometric modularity results. However, the proof as written does not verify one of the hypotheses of the quoted EHMM theorem for two of the five cases, so the main result is not yet fully established.

major comments (3)
  1. [Section 5, proof of Theorem 1, together with Theorem 2 condition (1)] For the cases (u,v)=(3,3) and (6,3), the cusp form fHD produced by the Euler integral is g1=K3(1/3,1)(3τ) and h1=K3(1/6,1)(6τ), respectively. Theorem 2 condition (1) requires T_p(fHD)=btilde_p·fHD for every prime p≡1 (mod M), but the paper never verifies this for g1 or h1. Indeed, Tables 2 and 3 record T2(g1)=-9g2 and T5(h1)=-81h5, showing that these functions are not Hecke eigenvectors. The eigenforms f27.3.b.b and f108.3.c.b are then constructed as linear combinations of the Galois orbit, but Theorem 2's hypothesis is about fHD, not about the eigenform completion. Unless the actual theorem in [4] has a weaker hypothesis, such as irreducibility of the Hecke module generated by the orbit, and that weaker version is stated and proved here, Theorem 2 does not apply to these two cases. This is a structural gap in the proof of Theorem 1.
  2. [Theorem 1 versus Theorem 2 and Proposition 2] Theorem 1 asserts the identity for every prime p≡1 (mod M), but Theorem 2 only supplies the conclusion for primes p>29 in the four cases handled by it, and Proposition 2 supplies p≥13 only for family (5). For families (1)–(4) the primes p=7,13,19 (and p=13 for family (3)) are not covered by any stated argument, and no numerical verification for these small primes is reported. The proof of Theorem 1 as stated is therefore incomplete; the small-prime cases must be checked explicitly, or Theorem 1 must be restricted to the range proven.
  3. [Example 1 and Tables 2–3] The Hecke operator actions are asserted with the phrase "It is straightforward to check" and no computational details are given. Since these actions uniquely determine the eigenform completions that appear in Theorem 1, the computations should be documented, for example by giving q-expansions up to the Sturm bound or by providing reproducible code.
minor comments (3)
  1. [Title] The title contains a typo: "DA WSEY" should be "DAWSEY".
  2. [Section 7, before Theorem 6] The symbol S2 is reused for the subgroup of squares in F_q^×, conflicting with the set S2 defined in (2.2); please rename one of the two objects.
  3. [Theorem 1 statement] The same symbol p is used for the rational prime and for the prime ideal above it; using a fraktur symbol such as \mathfrak p for the ideal would remove ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation is self-contained and the quoted EHMM is an external theorem.

full rationale

Theorem 1 is derived by applying the EHMM (Theorem 2, quoted from [4]) to the Euler-integral cusp forms K3(r,1)(Nτ). The target eigenforms are constructed from Galois orbits via explicit Hecke-operator computations (Tables 1–3, Example 1); no Hp value is used to fit the constants, and the non-unique sign choices are Galois-conjugate normalizations rather than free parameters matched to the right-hand side. Identities (3.14), Lemma 3, Corollary 1, and the Paley graph corollary follow by direct evaluation of Euler integrals, integration, and substitution of Theorem 1, not by assuming the conjectures. The reliance on the author's prior work [3,4] is a normal application of previously established theorems, with [4] published in Advances in Mathematics; the EHMM is not derived from the Dawsey–McCarthy conjectures being proved. One proof concern is that Remark 10 asserts condition (1) of Theorem 2 is satisfied for the Galois families, while Tables 2 and 3 show T2(g1) = -9g2 and T5(h1) = -81h5, so the individual K3 functions used as fHD are not Tp-eigenvectors; this is a correctness/hypothesis-checking gap in the proof as written, not a circular equivalence between the claimed output and its inputs, since the completed eigenforms are still computed without fitting any Hp data.

Assumptions & free parameters 0 free parameters · 7 assumptions · 0 invented entities

The central claim rests on prior theorems (EHMM, Katz, Ono, Borwein) rather than fitted parameters. The main internal assumptions are the asserted Hecke computations and the unstated small-prime verifications.

assumptions (7)
  • domain assumption Theorem 2 (EHMM criterion) from [4]
    The proof of Theorem 1 applies the general EHMM modularity criterion from the author's earlier work; the criterion is cited, not reproved, and carries the p > 29 bound.
  • standard math Katz's hypergeometric Galois representation theorem (Theorem 4)
    Provides the Galois representations whose traces are Hp and the Weil-type bounds used in Proposition 2 and Section 2.
  • standard math Ono's eta-quotient criterion (Theorem 3)
    Used to check that the K3 functions are holomorphic cusp forms of weight three with prescribed levels and Nebentypus.
  • domain assumption p^2 supercongruences (Theorem 2.3 of [4])
    Used in Proposition 2 to establish the mod p^2 divisibility that forces equality for family (5).
  • standard math Identity (4.2) for Hp functions from [19]
    Used to verify condition (2) of the EHMM for families (1)-(4); derived in Remark 16 from Greene's finite-field hypergeometric functions.
  • standard math Borwein cubic theta identities (3.9), (3.11)
    The three-term identity a^3 = b^3 + c^3 and the evaluation 2F1(1/3,2/3;1;t3)=a are used in the derivation of the K3 functions.
  • ad hoc to paper The asserted Hecke operator actions (T2, T3, T5, T7, T11)
    The eigenform completion in condition (1) of Theorem 2 rests on these computations, which are stated as 'straightforward' without details.

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Cite this review

Pith. "Pith review of On Some Hypergeometric Modularity Conjectures of Dawsey and McCarthy." pith.science (2026). https://pith.science/paper/KXJVQZPK

@misc{pith2026250719971,
  author       = {Pith},
  title        = {Pith review of: On Some Hypergeometric Modularity Conjectures of Dawsey and McCarthy},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KXJVQZPK}},
  note         = {Machine review of arXiv:2507.19971}
}
abstract

In recent work, the author, in collaboration with Allen, Long, and Tu, developed the Explicit Hypergeometric Modularity Method (EHMM), which establishes the modularity of a large class of hypergeometric Galois representations in dimensions two and three. One important application of the EHMM is the construction of an explicit family of eta-quotients, which we call the $\mathbb{K}_{2}$ functions, from the hypergeometric background. In this article, we introduce an analogous family of eta-quotients, which we call the $\mathbb{K}_{3}$ functions. These $\mathbb{K}_{3}$ functions are constructed using the theory of weight one cubic theta functions originally developed by Jonathan and Peter Borwein. We then use the $\mathbb{K}_{3}$ functions in the EHMM to resolve several hypergeometric modularity conjectures of Dawsey and McCarthy. Further, we provide applications to special $L$-values of the $\mathbb{K}_{3}$ functions and to the study of generalized Paley graphs.

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.