Pith. sign in

REVIEW 3 major objections 4 minor 50 references

The Explicit Hypergeometric Modularity Method III

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that hypergeometric Galois representations attached to four-parameter data κ(d,e) are explicitly modular: their Frobenius traces are exact Fourier coefficients of listed weight-4 Hecke eigenforms, plus a one-dimensional cy

desk verdict A solid, genuinely new variant of EHMM with a real gap where the proof determines the twist character; referee it, but require the omitted calculation. read the letter →

arxiv 2607.25173 v1 pith:KRONFUXU submitted 2026-07-28 math.NT

Brian Grove , Ling Long , Esme Rosen , Fang-Ting Tu This is my paper · ORCID
classification math.NT MSC 33C2011F0311F6611F8011T24
keywords hypergeometricfunctionsmodularformsGaloisrepresentationscharactersumsL-valuesCalabi-Yauthreefoldscommutativeformalgrouplawssupercongruences
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 refines the Explicit Hypergeometric Modularity Method so that it can handle hypergeometric data of length four whose field of definition is not Q. The central result is a precise bridge: for d=2,3,4 and e∈{2,3,4,6} with e dividing M_d, the semisimplified hypergeometric Galois representation attached to the datum κ(d,e) is isomorphic to the Galois representation of an explicitly listed weight-4 Hecke eigenform (possibly twisted by a quadratic character) plus a one-dimensional factor made of quadratic characters times the cyclotomic character. The proof shows the same congruence machinery works through a residue-theorem split of the datum, rather than the previous length-restricted decompositions, and extends to primes not necessarily congruent to 1 modulo the denominator. If the theorems are right, the Frobenius traces of these hypergeometric representations are computable as Fourier coefficients of explicit modular forms, and the nine associated rigid Calabi-Yau threefold modular forms come with closed q-expansions and period relations.

What carries the argument

The load-bearing object is the decomposition of the hypergeometric datum κ(d,e) (a pair of rational multisets of equal length) as κ_alg(1/e)⋆κ_3(d), together with the residue-theorem identity expressing F(κ(d,e),1) as a contour integral of the two factors. This splits the length-4 datum into a length-2 algebraic factor and a length-3 factor carrying the classical theory of elliptic functions to alternative bases, which provides modular parametrizations. The comparison is completed by commutative formal group laws (CFGL), giving p-adic congruences between the truncated hypergeometric series 4F3(...;1)_{p−1} and the p-th Fourier coefficient a_p(g_{d,e}) of the constructed weight-4 cusp form, w

What would settle it

Compute H(κ(3,2);1;p) directly from the character-sum definition for p=13 (p≡1 mod 6) and compare with ε_{3,2}(13)a_{13}(g_{3,2})+(−1)^{4+6}·13; any mismatch refutes Theorem 1.1 for that pair. Alternatively, verify the missing ±1 claim by computing (−C1(d)/4)^{(1−jp)/e}β_{d,e,j} mod p for each j∈(Z/eZ)^×, d∈{2,3,4}, e∈{2,3,4,6}; a single residue outside {+1,−1} would falsify the twist in Theorem 1.2.

Watch

Extended reading notes

Core claim

Central claim: for d∈{2,3,4}, e∈{2,3,4,6} with e|M_d, the semisimplified hypergeometric Galois representation of κ(d,e) is isomorphic to the ℓ-adic representation of an explicit weight-4 Hecke eigenform (twisted by χ_{−d} when d=3,4) plus a one-dimensional factor ς(d)ς(e)⊗ε_ℓ, where ς(e) is a quadratic character. Theorem 1.1 gives the matching trace identity: H(κ(d,e);1;p)=ε_{d,e}(p)a_p(g_{d,e})+(−1)^{(p−1)/d+(p−1)/e}p for all primes p≡1 mod lcm(d,e) with a_p(g_{d,e}) not divisible by p. The route: decompose the datum by a residue-theorem identity into a length-2 algebraic piece and a length-3 modular piece, then use commutative formal group laws to turn p-adic congruences between truncated

Load-bearing premise

Theorem 1.2 rests on the unproved claim that the p-adic expression (−C1(d)/4)^{(1−jp)/e}β_{d,e,j} is always ±1 mod p; the paper says 'explicit calculations show' this without displaying them, and if it ever took another value the asserted twist in the modularity isomorphism would be wrong.

Editorial extensions

If this is right

  • The Galois representation attached to κ(d,e) has its Frobenius traces computable as Fourier coefficients of a known weight-4 modular form, for d=2,3,4 and e∈{2,3,4,6}.
  • The trace identity of Theorem 1.1 gives a deterministic way to evaluate the finite-field hypergeometric function H(κ(d,e);1;p) for all primes p≡1 mod lcm(d,e) with a_p(g_{d,e}) not divisible by p.
  • Nine rigid Calabi-Yau threefold modular forms are produced in closed form as eta-quotient/Eisenstein combinations, so their q-expansions, Hecke eigenvalues, and special L-values can be computed directly.
  • The residue-theorem variant extends the method to augmenting data of length greater than 1 and to primes not congruent to 1 modulo the common denominator, removing two restrictions of the earlier approach.
  • For CM forms in the constructed families, explicit period relations tie special L-values to gamma quotients, giving concrete instances of the expected algebraicity for hypergeometric motive L-values.

Reading between the lines

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

  • The same residue-theorem split should apply to other length-four data, not just κ(d,e), since the only datum-specific input is the pair {r,1−r} in the augmenting factor; this suggests a general recipe for length-four EHMM and for higher length via iterated splits.
  • The explicit character calculation flagged in the proof of Theorem 1.2 could be turned into a constructive criterion: for any divisor e, the sign of (−C1(d)/4)^{(1−jp)/e}β_{d,e,j} mod p determines the twist, so verifying the ±1 assertion for all j,p would also provide a fast check of the isomorphism for new e.
  • The period identities derived in Section 6 suggest a direct route from the method to Deligne-style period relations: the same contour integral that gives modularity also expresses hypergeometric evaluations as integrals of modular forms, so special L-values of the constructed forms can be computed without invoking general period conjectures.
  • Because Theorem 1.1 also covers e=12, where the datum is defined over a totally real subfield rather than Q, the method may connect to Hilbert modular forms for data with non-trivial stabilizer fields.
Share X Bluesky LinkedIn Reddit HN

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 develops a variant of the authors' Explicit Hypergeometric Modularity Method, based on Zagier's residue-trick identity (1.1), and applies it to the length-four data κ(d,e) = {{1/d,(d−1)/d,1/e,(e−1)/e},{1,1,1,1}} for d=2,3,4 and e dividing M_d. It constructs explicit weight-4 cusp-form spaces G_{d,e} (Theorem 1.1) and, for e∈{2,3,4,6}, states an explicit isomorphism (Theorem 1.2) between the Beukers–Cohen–Mellit hypergeometric Galois representation and ρ_{f^#_{d,e}} plus a cyclotomic summand. The method is also used to give explicit constructions for modular forms attached to the nine Rodriguez-Villegas rigid Calabi-Yau threefolds previously treated by Long–Tu–Yui–Zudilin [35], and to compute some L-values.

Significance. If the gaps identified below are repaired, this would be a valuable uniform, explicit method: it gives concrete q-expansions and level/character data for the weight-4 eigenforms, and it connects CFGL congruences, hypergeometric supercongruences from [4], and Galois representations without relying on database searches. The central modularity statement for e∈{2,3,4,6} is independently supported by [35], which increases confidence in the result. The paper also offers interesting L-value identities and period relations. However, as written, a load-bearing quadratic-twist determination is omitted, and the p-adic setup used for the CFGL argument needs further justification for some allowed primes.

major comments (3)
  1. [§5.3, proof of Theorem 1.2 and footnote 1] The exact isomorphism in Theorem 1.2 is decided by the finite-order character that compares ρ_{f^#_{d,e}}|_{G_{Q(ζ_e)}} with ρ_{g_{d,e}}|_{G_{Q(ζ_e)}}. The proof states that this character is determined by (−C1(d)/4)^{(1−jp)/e} β_{d,e,j} (mod p), and footnote 1 admits 'It is unclear otherwise why (−C1(d)/4)^{(1−jp)/e} β_{d,e,j} ≡ ±1 (mod p)' before asserting 'Explicit calculations show ...' without giving them. This is not a routine detail: if the character were a different quadratic character, the two-dimensional summand would be a different twist and the displayed decomposition in Theorem 1.2 would be false. Please provide the actual calculation, or a theorem verifying the claim for every (d,e) in the list, including e=6.
  2. [§3, Proposition 3.1 and Remark 3.1(1)] Proposition 3.1 asserts that the G_{d,i/M_d}(M_d τ) are congruence cusp forms spanning a Hecke-invariant space, and that the tabulated β-combinations are Hecke eigenforms with β^2_{d,e,i}∈Z. Remark 3.1(1) says the details are omitted but 'checked case-by-case'. This is load-bearing: Proposition 4.3 and Lemma 4.4 require both the Hecke recursion for g_{d,e} and β_{d,e,j}∈R^×. The paper should either include the case-by-case verification (as is done for G_{3,3}) or give precise references that contain it.
  3. [§4.2, Eq. (4.3) and Lemma 4.4] The CFGL theorem (Theorem 4.2) requires a ring automorphism σ of R=Z_p[C1(d)^{1/e}, β_{d,e,j}] with σ(x)≡x^p (mod pR). The existence of such σ is not automatic for the primes allowed by the paper, since the assumption is only p∤de. For example, in the G_{4,12} family (Table 5) the coefficients involve √13; at p=13, which satisfies p∤48 and p≡1 (mod 12), Z_13[√13] is ramified and admits no automorphism σ with σ(x)≡x^p mod 13. Similar ramification issues can arise for other families. The proof needs either an explicit lemma establishing the existence of σ for all p used in the argument, or a restriction of the statement to primes unramified in the relevant fields, with the remaining primes handled separately.
minor comments (4)
  1. [§2 (paragraph before §2.1)] Typo: 'expected to to be the étale realization' should be 'expected to be the étale realization'.
  2. [Table 5, G_{4,3} row] The row lists 'f9.4.a.a(τ)=G4(1/3)(3τ)' and 'f9.4.a.a(2τ)=G4(2/3)(3τ)' but the Level column says 18. Using the LMFDB label f9.4.a.a for a level-18 form is confusing; please clarify the level and the relation between the two forms.
  3. [§2.3] The outline says the supercongruence from [4] applies when p≡1 (mod M(d,e)), while Theorem 1.1 is stated for p≡1 (mod lcm(d,e)). Please align the notation and state the exact congruence modulus used.
  4. [§5.1] Proposition 5.1 cites [9,26] but the decomposition (5.1) with φ_{d,e} finite order is asserted without proof. Please state clearly where this decomposition is proved or give the argument; it is used in the proof of Theorem 1.2.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular reduction: the trace identity and representation isomorphism are derived from external supercongruences, CFGL congruences, and Chebotarev; the flagged footnote is a completeness gap, not a circular step.

full rationale

The central claims are not circular. Theorem 1.1's trace identity is not definitional: H(κ(d,e);1;p) is an independent Gauss-sum trace, ε_{d,e}(p) is defined by a congruence, and the equality is reached through [4, Thm 2.3], the CFGL congruences of Proposition 4.3, Katz's bound |H| ≤ 3p^{3/2}, the Weil bound, and the mod p^2-to-equality upgrade. The spaces G_{d,e} are constructed from REAB/Clausen data rather than fitted to the target traces; the β_{d,e,j} are chosen to make Hecke eigenforms, not to force the theorem. Theorem 1.2 likewise uses Theorem 1.1, Chebotarev, and the independent Faltings–Serre-based result [35], with the final quadratic twist checked against LMFDB data. The paper does contain a genuine omitted verification rather than a circular one: §5.3, footnote 1 admits 'It is unclear otherwise why (−C1(d)/4)^{(1−jp)/e} β_{d,e,j} ≡ ±1 (mod p)' and says 'Explicit calculations show ...' without displaying them; if those calculations did not yield the trivial character or χ_{−d}, the stated isomorphism in Theorem 1.2 would fail. Similarly, Remark 3.1(1) omits the case-by-case verification of Proposition 3.1. These are missing proofs, not assumptions of the theorem, so they do not make the derivation circular. The heavy self-citations, especially [4] and [30], are load-bearing but are published or independently grounded results with stated assumptions that do not include the target theorem; hence they do not constitute circularity. Score 2 reflects the conservative weight of self-cited inputs and the two flagged omitted checks, not a reduction of the result to its own inputs.

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

The central claim rests on a network of prior theorems from the authors' own EHMM program and from Katz/BCM/Stienstra-Beukers. No numbers are fitted to the hypergeometric trace data: d and e are inputs, C1(d) are fixed leading q-coefficients of Hauptmoduln, beta_{d,e,i} are determined by the Hecke eigenform condition and tabulated, and epsilon_{d,e}(p) is defined by a congruence. The paper introduces no new particles, forces, or ad hoc geometric objects.

assumptions (7)
  • domain assumption Theorem 2.3 of [4]: modulo-p^2 supercongruence identifying H(kappa;1;F_p) with truncated hypergeometric F(kappa,1)_{p-1} up to an explicit sign/normalization.
    Invoked in Section 2.3 and Sections 5.1-5.2 as the bridge from character sums to truncated series; cited from EHMM I and not reproved here.
  • standard math Stienstra-Beukers CFGL comparison [46, Thm A.9] between u-expansion and q-expansion coefficients.
    Used in Lemma 4.4 and Proposition 4.3 to convert Hecke recursions into congruences for truncated hypergeometric coefficients.
  • domain assumption Katz [26] / Beukers-Cohen-Mellit [9] existence of eta_{kappa,ell,1} and its BCM extension with Frobenius traces given by H.
    Proposition 5.1 is quoted from these references; no proof is given in this paper.
  • domain assumption Dwork unit root and etale-crystalline comparison make the unit root congruent to the trace modulo p.
    Section 2 states this is 'anticipated' from comparison isomorphisms; the paper relies on it for (5.3)-(5.5).
  • ad hoc to paper Proposition 3.1: G_d(i/M_d)(M_d tau) are congruence cusp forms spanning a Hecke-invariant space, with the tabulated beta-combinations as eigenforms.
    Proof omitted; Remark 3.1(1) says details have been checked case-by-case but are not included.
  • ad hoc to paper The finite-order character in Theorem 1.2 is trivial or chi_{-d}; determined by 'explicit calculations' not shown.
    Footnote 1 in Section 5.3 explicitly says it is unclear why the congruence yields ±1; the conclusion rests on unstated computations.
  • ad hoc to paper Technical p-adic setup: R = Z_p[C1(d)^{1/e}, beta_{d,e,j}] has an automorphism sigma with sigma(a) ≡ a^p mod pR and beta_{d,e,j} in R^×.
    Assumed in Section 4.2 for all primes p not dividing de; not proven for the non-rational beta's appearing in the e=8,12,24 tables.

how reviews work

0 comments
Cite this review

Pith. "Pith review of The Explicit Hypergeometric Modularity Method III." pith.science (2026). https://pith.science/paper/KRONFUXU

@misc{pith2026260725173,
  author       = {Pith},
  title        = {Pith review of: The Explicit Hypergeometric Modularity Method III},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/KRONFUXU}},
  note         = {Machine review of arXiv:2607.25173}
}
abstract

We refine the Explicit Hypergeometric Modularity Method (EHMM) and develop a variant that applies to a broader class of hypergeometric data. As an application, we establish the modularity of hypergeometric Galois representations arising from length-$4$ data that are not necessarily defined over $\mathbb{Q}$. We also use this method to give explicit constructions of nine modular forms associated with hypergeometric rigid Calabi-Yau threefolds conjectured to be modular by Rodriguez-Villegas. The modularity of these threefolds was first proved by Long-Tu-Yui-Zudilin using a different approach based on the Faltings-Serre method. Moreover, the explicit nature of the method makes it well suited to the computation of special $L$-values of the associated modular forms.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

50 extracted references · 3 linked inside Pith

  1. [35]

    Supercongruences for rigid hyper- geometric Calabi-Yau threefolds.Adv

    Ling Long, Fang-Ting Tu, Noriko Yui, and Wadim Zudilin. Supercongruences for rigid hyper- geometric Calabi-Yau threefolds.Adv. Math., 393:Paper No. 108058, 49, 2021. 34

    2021

  2. [4]

    The Explicit Hypergeometric- Modularity Method I.Adv

    Michael Allen, Brian Grove, Ling Long, and Fang-Ting Tu. The Explicit Hypergeometric- Modularity Method I.Adv. Math., 487, Paper No. 110411, 2025

    2025

  3. [1]

    Stegun, editors.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, volume 55 ofNational Bureau of Standards Ap- plied Mathematics Series

    Milton Abramowitz and Irene A. Stegun, editors.Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, volume 55 ofNational Bureau of Standards Ap- plied Mathematics Series. U.S. Government Printing Office, Washington, D.C., 1964

    1964

  4. [2]

    Modularity of a certain Calabi-Yau threefold.Monatsh

    Scott Ahlgren and Ken Ono. Modularity of a certain Calabi-Yau threefold.Monatsh. Math., 129(3):177–190, 2000

    2000

  5. [3]

    The EHMM Calcu- lator.https://sites.google.com/view/esme-rosen/ehmm-calculator, 2026

    Michael Allen, Brian Grove, Ling Long, Esme Rosen, and Fang-Ting Tu. The EHMM Calcu- lator.https://sites.google.com/view/esme-rosen/ehmm-calculator, 2026. [Online; ac- cessed 24 July 2026]

    2026

  6. [5]

    The Explicit Hypergeometric- Modularity Method II.Research in the Mathematical Sciences, 12, Paper No

    Michael Allen, Brian Grove, Ling Long, and Fang-Ting Tu. The Explicit Hypergeometric- Modularity Method II.Research in the Mathematical Sciences, 12, Paper No. 84, 2025

    2025

  7. [6]

    Andrews, Richard Askey, and Ranjan Roy.Special functions, volume 71 ofEncyclo- pedia of Mathematics and its Applications

    George E. Andrews, Richard Askey, and Ranjan Roy.Special functions, volume 71 ofEncyclo- pedia of Mathematics and its Applications. Cambridge University Press, Cambridge, 1999

    1999

  8. [7]

    Explicit hypergeometric modularity of certain weight two and four Hecke eigenforms, 2026, arXiv 2604.02723

    Rupam Barman and Sipra Maity. Explicit hypergeometric modularity of certain weight two and four Hecke eigenforms, 2026, arXiv 2604.02723

    arXiv 2026

Show all 50 references
  1. [8]

    Berndt, S

    Bruce C. Berndt, S. Bhargava, and Frank G. Garvan. Ramanujan’s theories of elliptic functions to alternative bases.Trans. Amer. Math. Soc., 347(11):4163–4244, 1995

    1995

  2. [9]

    Finite hypergeometric functions.Pure Appl

    Frits Beukers, Henri Cohen, and Anton Mellit. Finite hypergeometric functions.Pure Appl. Math. Q., 11(4):559–589, 2015

    2015

  3. [10]

    J. M. Borwein and P. B. Borwein. A Cubic Counterpart of Jacobi’s Identity and the AGM. Transactions of the American Mathematical Society, 323(2):691–701, 1991

    1991

  4. [11]

    Borwein and Peter B

    Jonathan M. Borwein and Peter B. Borwein.Pi and the AGM, volume 4 ofCanadian Math- ematical Society Series of Monographs and Advanced Texts. John Wiley & Sons, Inc., New York, 1998. A study in analytic number theory and computational complexity, Reprint of the 1987 original, A...

    1998

  5. [12]

    Borwein, Peter B

    Jonathan M. Borwein, Peter B. Borwein, and Frank. G. Garvan. Some cubic modular identities of Ramanujan.Trans. Amer. Math. Soc., 343(1):35–47, 1994

    1994

  6. [13]

    American Mathematical Society, Providence, RI, 2017

    Henri Cohen and Fredrik Str¨ omberg.Modular forms, volume 179 ofGraduate Studies in Math- ematics. American Mathematical Society, Providence, RI, 2017. A classical approach

    2017

  7. [14]

    P. Deligne. Valeurs de fonctionsLet p´ eriodes d’int´ egrales. InAutomorphic forms, representa- tions andL-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), 33 Part 2, volume XXXIII ofProc. Sympos. Pure Math., pages 313–346. Amer. Math. Soc., Pro...

    1977

  8. [15]

    Raghuram

    Pierre Deligne and A. Raghuram. Motives, periods, and functoriality.Tunis. J. Math., 7(1):131–165, 2025

    2025

  9. [16]

    Hautes ´Etudes Sci

    Bernard Dwork.p-adic cycles.Inst. Hautes ´Etudes Sci. Publ. Math., pages 27–115, 1969

    1969

  10. [17]

    Variations of Hodge structures for hypergeometric differential operators and parabolic Higgs bundles.Int

    Roman Fedorov. Variations of Hodge structures for hypergeometric differential operators and parabolic Higgs bundles.Int. Math. Res. Not. IMRN, pages 5583–5608, 2018

    2018

  11. [18]

    Ann., 387(3-4):1629–1689, 2023

    Lei Fu, Peigen Li, Daqing Wan, and Hao Zhang.p-adic GKZ hypergeometric complex.Math. Ann., 387(3-4):1629–1689, 2023

    2023

  12. [19]

    Hypergeo- metric functions over finite fields.Mem

    Jenny Fuselier, Ling Long, Ravi Ramakrishna, Holly Swisher, and Fang-Ting Tu. Hypergeo- metric functions over finite fields.Mem. Amer. Math. Soc., 280(1382), 2022

    2022

  13. [20]

    Fuselier and Dermot McCarthy

    Jenny G. Fuselier and Dermot McCarthy. Hypergeometric type identities in thep-adic setting and modular forms.Proc. Amer. Math. Soc., 144(4):1493–1508, 2016

    2016

  14. [21]

    Benedict H. Gross. On the periods of abelian integrals and a formula of Chowla and Selberg. Invent. Math., 45(2):193–211, 1978. With an appendix by David E. Rohrlich

    1978

  15. [22]

    On Some Hypergeometric Modularity Conjectures of Dawsey and McCarthy, 2025, arXiv 2507.19971

    Brian Grove. On Some Hypergeometric Modularity Conjectures of Dawsey and McCarthy, 2025, arXiv 2507.19971

    2025 arXiv

  16. [23]

    Hypergeometric local systems overQwith Hodge vector (1,1,1,1), 2024, arXiv 2401.13529

    Giulia Gugiatti and Fernando Rodriguez Villegas. Hypergeometric local systems overQwith Hodge vector (1,1,1,1), 2024, arXiv 2401.13529

    2024 arXiv

  17. [24]

    Hypergeometric evaluations ofL-values of weakly holomorphic modular forms and harmonic maass forms

    Che-Wei Hsu, Fang-Ting Tu, and Yifan Yang. Hypergeometric evaluations ofL-values of weakly holomorphic modular forms and harmonic maass forms. In preparation

  18. [25]

    Nicholas M. Katz. Travaux de Dwork. InS´ eminaire Bourbaki, 24` eme ann´ ee (1971/1972), Exp. No. 409, pages 167–200. Lecture Notes in Math., Vol. 317. Springer, 1973

    1971

  19. [26]

    Katz.Exponential sums and differential equations, volume 124 ofAnnals of Math- ematics Studies

    Nicholas M. Katz.Exponential sums and differential equations, volume 124 ofAnnals of Math- ematics Studies. Princeton University Press, Princeton, NJ, 1990

    1990

  20. [27]

    Katz.Rigid local systems, volume 139 ofAnnals of Mathematics Studies

    Nicholas M. Katz.Rigid local systems, volume 139 ofAnnals of Mathematics Studies. Princeton University Press, Princeton, NJ, 1996

    1996

  21. [28]

    Kiran S. Kedlaya. Frobenius structures on hypergeometric equations. InArithmetic, geometry, cryptography, and coding theory 2021, volume 779 ofContemp. Math., pages 133–158. Amer. Math. Soc., [Providence], RI, [2022]©2022

    2021

  22. [29]

    Trace formulas for Hecke operators, Gaussian hypergeometric functions, and the modularity of a threefold.J

    Catherine Lennon. Trace formulas for Hecke operators, Gaussian hypergeometric functions, and the modularity of a threefold.J. Number Theory, 131(12):2320–2351, 2011

    2011

  23. [30]

    The arithmetic of hypergeometric Galois representations, preprint 2026

    Wen-Ching Winnie Li, Tong Liu, and Ling Long. The arithmetic of hypergeometric Galois representations, preprint 2026

    2026

  24. [31]

    Computing specialL-values of certain modular forms with complex multiplication.SIGMA 14 (2018), 090, August 2018

    Wen-Ching Winnie Li, Ling Long, and Fang-Ting Tu. Computing specialL-values of certain modular forms with complex multiplication.SIGMA 14 (2018), 090, August 2018

    2018

  25. [32]

    A Whipple7F6 formula revisited.Matem- atica, 1(2):480–530, 2022

    Wen-Ching Winnie Li, Ling Long, and Fang-Ting Tu. A Whipple7F6 formula revisited.Matem- atica, 1(2):480–530, 2022

    2022

  26. [33]

    Some numeric hypergeometric supercongruences

    Ling Long. Some numeric hypergeometric supercongruences. InVertex operator algebras, num- ber theory and related topics, volume 753 ofContemp. Math., pages 139–156. Amer. Math. Soc., Providence, RI, 2020

    2020

  27. [34]

    Some supercongruences occurring in truncated hypergeo- metric series.Adv

    Ling Long and Ravi Ramakrishna. Some supercongruences occurring in truncated hypergeo- metric series.Adv. Math., 290:773–808, 2016

    2016

  28. [36]

    Transformations of well-poised hypergeometric functions over finite fields

    Dermot McCarthy. Transformations of well-poised hypergeometric functions over finite fields. Finite Fields Appl., 18(6):1133–1147, 2012

    2012

  29. [37]

    Papanikolas

    Dermot McCarthy and Matthew A. Papanikolas. A finite field hypergeometric function asso- ciated to eigenvalues of a Siegel eigenform.Int. J. Number Theory, 11(8):2431–2450, 2015

    2015

  30. [38]

    Ramanujan

    S. Ramanujan. Modular equations and approximations toπ[Quart. J. Math.45(1914), 350– 372]. InCollected papers of Srinivasa Ramanujan, pages 23–39. AMS Chelsea Publ., Provi- dence, RI, 2000

    1914

  31. [39]

    Roberts and Fernando Rodriguez Villegas

    David P. Roberts and Fernando Rodriguez Villegas. Hypergeometric motives.Notices Amer. Math. Soc., 69(6):914–929, 2022

    2022

  32. [40]

    Hypergeometric families of Calabi-Yau manifolds

    Fernando Rodriguez-Villegas. Hypergeometric families of Calabi-Yau manifolds. InCalabi-Yau varieties and mirror symmetry (Toronto, ON, 2001), volume 38 ofFields Inst. Commun., pages 223–231. Amer. Math. Soc., Providence, RI, 2003

    2001

  33. [41]

    Rogers, J

    M. Rogers, J. G. Wan, and I. J. Zucker. Moments of elliptic integrals and criticalL-values. Ramanujan J., 37(1):113–130, 2015

    2015

  34. [42]

    Explicit modularity of reducible rank 2 hypergeometric motives I

    Esme Rosen. Explicit modularity of reducible rank 2 hypergeometric motives I. In preparation

  35. [43]

    Esme Rosen.L-values of certain weight 3 modular forms and transformations of hypergeometric series.Res Math Sci, 13(57), 2026

    2026

  36. [44]

    Modular Forms and Certain 2F1(1) Hypergeometric Series.Proc

    Esme Rosen. Modular Forms and Certain 2F1(1) Hypergeometric Series.Proc. of the Amer. Math. Soc., 154(7):2803–2818, 2026

    2026

  37. [45]

    The special values of the zeta functions associated with cusp forms.Comm

    Goro Shimura. The special values of the zeta functions associated with cusp forms.Comm. Pure Appl. Math., 29(6):783–804, 1976

    1976

  38. [46]

    On the Picard-Fuchs equation and the formal Brauer group of certain ellipticK3-surfaces.Math

    Jan Stienstra and Frits Beukers. On the Picard-Fuchs equation and the formal Brauer group of certain ellipticK3-surfaces.Math. Ann., 271(2):269–304, 1985

    1985

  39. [47]

    P. F. Stiller. Classical automorphic forms and hypergeometric functions.J. Number Theory, 28(2):219–232, 1988

    1988

  40. [48]

    On differential equations satisfied by modular forms.Math

    Yifan Yang. On differential equations satisfied by modular forms.Math. Z., 246(1-2):1–19, 2004

    2004

  41. [49]

    The arithmetic and topology of differential equations

    Don Zagier. The arithmetic and topology of differential equations. InEuropean Congress of Mathematics, pages 717–776. Eur. Math. Soc., Z¨ urich, 2018

    2018

  42. [50]

    A hypergeometric version of the modularity of rigid Calabi-Yau manifolds

    Wadim Zudilin. A hypergeometric version of the modularity of rigid Calabi-Yau manifolds. SIGMA Symmetry Integrability Geom. Methods Appl., 14:Paper No. 086, 16, 2018. Department of Mathematics, West Virginia University Institute of Technology, Beckley, WV 25801, USA Email addr...

    2018

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.