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Explicit classes in Habiro cohomology

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Pith's one-line read Explicit classes in Habiro cohomology are constructed from q-deformations of hypergeometric motives, giving q-Picard-Fuchs equations that link quantum K-theory and complex Chern-Simons theory.

desk verdict A genuinely creative construction with a load-bearing gap: the main theorem's classes are meromorphic with poles at roots of unity, while the cohomology module as defined only admits convergent ordinary power series. read the letter →

arxiv 2505.19885 v1 pith:FDPA4QJF submitted 2025-05-26 math.AG hep-thmath.GT

Stavros Garoufalidis , Campbell Wheeler This is my paper · ORCID
classification math.AGhep-thmath.GT MSC 14F4014D0714J3233D15
keywords Habirocohomologyringhypergeometricmotivesq-Picard-Fuchsequationq-holonomicmodulesCalabi-YaufamiliescomplexChern-SimonstheoryquantumK-theory
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 constructs explicit cohomology classes in a newly defined “naive” Habiro cohomology for smooth varieties over étale arithmetic rings. It defines $H^n_{\mathrm{naiv}}(X/B)$ by expanding relative algebraic de Rham cohomology near every root of unity and gluing the expansions through Frobenius twists. The main theorems show that q-deformations of hypergeometric motives produce collections $\omega_q = (\omega_{m,q-\zeta_m})$ that lie in this cohomology and are annihilated by q-Picard-Fuchs equations. The same framework, via a push-forward from the Habiro ring, covers the Legendre elliptic family, the A-polynomial curve of the figure-eight knot, and the quintic threefold, so quantum K-theory and complex Chern-Simons theory appear as instances of one construction.

What carries the argument

The central object is the naive Habiro cohomology $H^n_{\mathrm{naiv}}(X/B) = H(H^n_{\mathrm{dR}}(X/B))$ of Definition 1.2: a collection of series in $q-\zeta_m$ valued in relative algebraic de Rham cohomology, p-adically convergent and glued by the Frobenius endomorphism with a $\lambda \mapsto \lambda^{1/m}$ twist. The second ingredient is the q-deformation of a hypergeometric motive: replacing Pochhammer symbols by q-Pochhammer symbols turns $_{n+1}F_n$ into its basic hypergeometric counterpart $_{n+1}\varphi_n$, and expansions of q-Pochhammer symbols near roots of unity produce the differential operators $D_m(\lambda^{1/m},\theta,q-\zeta_m)$ and q-Beta factors $B_m$ that convert the classical solution into a cohomology class. The push-forward method uses the Habiro ring of an étale algebra together with the residue identity $\mathrm{Res}_{x=0} \approx x$-constant term to map Habiro-ring elements to classes on $X/B$.

What would settle it

For the Legendre family of Section 2.6, compute the series $\omega_{m,q-\zeta_m}$ of Equation (26) to high order, re-expand at a prime root $\zeta_{pm}$, apply the Frobenius twist, and test the gluing equality (12) modulo $p^N$ for increasing $N$; the first mismatch would disprove the claim that $\omega_q\in H^n_{\mathrm{naiv}}(X/B)$. The paper's own $p=7$ check on the figure-eight A-polynomial curve shows the type of numerical test that is within reach.

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Extended reading notes

Core claim

The paper establishes that explicit nontrivial cycles in Habiro cohomology exist and can be written down. For a hypergeometric motive with vectors $\alpha,\beta$, the q-Pochhammer deformation $_{n+1}\varphi_n$ of the hypergeometric series, expanded near each root of unity and normalized by q-Beta factors, defines a collection $\omega_q$ in $H^n_{\mathrm{naiv}}(X/B)$ (Theorem 1.5); this class generates a q-holonomic submodule annihilated by the q-Picard-Fuchs equation (98) (Theorem 1.6). A complementary method pushes forward elements of the Habiro ring of an étale algebra, such as symmetrized Nahm sums, into Habiro cohomology (Theorem 1.18). The three worked examples are the Legendre family, the A-polynomial curve of the figure-eight knot, and the quintic threefold, whose q-Picard-Fuchs equation had already appeared in genus-zero quantum K-theory.

Load-bearing premise

The load-bearing premise is that the newly defined “naive” Habiro cohomology captures the actual Habiro cohomology, an expectation stated by the authors, together with the unpublished finiteness and base-change input used to prove Theorem 1.5; if either gives way, the main theorems concern a different object.

Editorial extensions

If this is right

  • Every hypergeometric motive yields explicit classes $\omega_q$ in $H^n_{\mathrm{naiv}}(X/B)$, not merely an abstract existence statement.
  • The classes generate q-holonomic submodules, giving q-difference equations that reduce to the classical Picard-Fuchs equation at $q=1$.
  • One-parameter Calabi-Yau families, including the quintic threefold, carry explicit Habiro cohomology classes whose q-Picard-Fuchs equations reproduce those of genus-zero quantum K-theory.
  • Symmetrized Nahm sums push forward to Habiro cohomology classes, so knot-theoretic curves such as the A-polynomial of the figure-eight knot enter the same framework as complex Chern-Simons theory.

Reading between the lines

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

  • If the naive cohomology is later identified with the full Habiro cohomology, these explicit classes would provide concrete generators for that module, and the different q-holonomic ranks found for two Legendre deformations suggest Habiro cohomology contains quantum directions invisible to classical de Rham cohomology.
  • The constant-term formula of Remark 1.9 predicts Hasse polynomials modulo primes; computing those constant terms for hypergeometric families beyond the paper's examples would test the geometric content of the construction.
  • If the q-Borel twist of Question 1.17 descends to Habiro cohomology, it would produce families of classes indexed by symmetric matrices and could connect the construction to the 3D-index of hyperbolic knots.
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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

4 major / 6 minor

Summary. The paper proposes an explicit cycle description of Habiro cohomology through a newly defined 'naive' object H^n_naiv(X/B): collections of p-adically convergent power series around all roots of unity, valued in algebraic de Rham cohomology and glued by Frobenius twists. The central theorems construct explicit classes in H^n_naiv(X/B) from q-deformations of hypergeometric motives (Theorems 1.5 and 1.6), from push-forwards of Habiro ring elements (Theorems 1.11, 1.13, 1.18, 1.19), and for the A-polynomial curve of the figure-eight knot (Theorem 4.5). The paper is illustrated with the Legendre family, the figure-eight knot curve, and the quintic threefold, and it argues that these constructions unify quantum K-theory and complex Chern–Simons theory around higher-dimensional critical loci.

Significance. If the theorems are correct, the paper provides the first explicit nontrivial cycles in a Habiro-type cohomology, connecting q-holonomic modules and q-deformed Picard–Fuchs equations to quantum K-theory and to the 3D-index of knots. The manuscript contains substantial original constructions, detailed proofs for many auxiliary statements, and a large amount of explicit computational evidence. Particular strengths are the independent definition of the naive Habiro cohomology, the explicit B_m normalization linked to p-adic gamma functions, and the concrete verification of Frobenius gluing in the Fermat-curve case. However, the main theorems are proved only for the newly invented H^n_naiv, and the advertised identification with Wagner–Scholze Habiro cohomology is explicitly only an expectation; moreover, as discussed below, one load-bearing compatibility issue affects the central theorem even for the naive theory.

major comments (4)
  1. [§1.3, Eq. (11); §2.5, Theorem 2.9; Eq. (26)] Definition 1.2 requires the components f_m(q−ζ_m) to be ordinary convergent power series in V_m⊗Q[ζ_m][[q−ζ_m]]. The class ω_{m,q−ζ_m} whose membership is the content of Theorem 1.5 is defined in Eq. (26) to live in (q−ζ_m)^{−n−1}Q[ζ_m][[q−ζ_m]], and each factor B_m has a simple pole at q=ζ_m. The convergence proof in Theorem 2.9 establishes only that log(q)^{n+1}D_m^rem is convergent. Since log(q) vanishes at every root of unity in C_p, it is not invertible in the ring of convergent power series on |q−ζ_m|_p<1, so the displayed poles do not disappear. Thus Theorem 1.5 is not a formal consequence of the stated definitions. The authors must either prove that the pole contributions cancel after the D_m action, or amend Definition 1.2 and the gluing condition (12) to allow meromorphic series with bounded pole order and then re-verify the Habiro-module structure. This is an internal compatibility issue and is load-bearing for the main claim.
  2. [§2.5, proof of Theorem 1.5] The proof of Theorem 1.5 depends on the statement that H^n_dR(X/B) is finitely generated, 'as was kindly communicated to us by P. Scholze and F. Wagner', and then on a base-change injection into H^n_dR(X/B)⊗Q((λ)). This is unpublished external input. Because the gluing equality is deduced from an equality in the λ-completed tensor product, the theorem is conditional on this communicated result. The paper should either prove the finite-generation and base-change facts, or state them explicitly as assumptions in the theorem. As written, a referee cannot verify the central claim from the manuscript alone.
  3. [§1.2, §1.3; abstract; Theorems 1.5, 1.6] The abstract and introduction advertise results about 'Habiro cohomology', but all proofs target the newly introduced H^n_naiv(X/B). Section 1.3 explicitly says 'We expect that this definition should capture some important features of Habiro cohomology being developed by Wagner and Scholze.' Thus the equivalence to the actual Habiro cohomology is a conjecture, and all theorems are conditional on it. This is not an internal inconsistency, but it changes the advertised significance and should be stated prominently in the abstract and in the theorem statements, for example by saying 'for the naive Habiro cohomology' throughout.
  4. [§2.8, Eq. (153); §4.6, Eq. (251); §2.6, Eq. (135)] Several of the paper's applications rely on large computer computations: the 24th-order q-difference equation for the quintic, the 6th-order equation for the figure-eight elliptic family, the 150-coefficient expansions, and the qHolonomic-generated operators. These computations are presented without the scripts, data files, or a reproducibility statement. Since Theorem 1.6 and several exhibited q-Picard–Fuchs deformations depend on such computations, the reader cannot independently check them. The authors should either ship the code and output, or clearly separate rigorously proved statements from computer-assisted claims.
minor comments (6)
  1. [Abstract] There is a typo: 'Lege ndre' should be 'Legendre'.
  2. [§1.5 heading] 'explcit elements' should read 'explicit elements'.
  3. [Throughout] The hyphenation of 'q-Picard Fuchs' is inconsistent; it should be 'q-Picard–Fuchs' everywhere.
  4. [§2.5, Theorem 2.9] The phrase 'We can write' in the proof is a grammatical fragment; the proof would benefit from a short explanation of why right division by P yields a remainder of degree at most n, even though the earlier statement says 'n+1 is the order of P'.
  5. [§1.3, Remark 1.3] The sentence 'It is curious that the logarithmic-growth of the powers of q−ζ_m matches the Christol-Dwork-Robba bounds...' is vague; a precise statement or reference would help the reader understand the role of logarithmic growth in the convergence conditions.
  6. [§2.2, Corollary 2.5] The notation H^1_naiv(X_N/Z[1/(2N)!]) uses a base that is a localization of Z; the paper elsewhere uses B=Spec(R) with R/Z[λ] étale, and it would be helpful to explain how this base fits into Definition 1.2.

Circularity Check

1 steps flagged · score 2.0 of 10

No circular fit in the main hypergeometric construction; only a minor self-citation in the push-forward of Habiro-ring elements.

  1. self citation load bearing [Section 3.2, Proof of Theorem 1.13]
    "Our goal is to give a self-contained proof of the following theorem implicit from [17]. ... The other inclusion is proven as in [17, Eqn.(169)]."

    Theorem 1.13 is an input to the push-forward construction of Habiro-cohomology classes (Theorems 1.12 and 1.18). The theorem is explicitly attributed to the authors' own previous work [17], and the proof described as self-contained delegates the nontrivial inclusion R⊗Q∩Z[[t]]=R to [17, Eqn.(169)] rather than proving it. Thus the push-forward examples rely on a prior paper by the same authors for a load-bearing inclusion. This is minor because the main hypergeometric construction (Theorem 1.5) is proved independently from q-Pochhammer expansions and de Rham base change, not from [17].

full rationale

The central theorem (Theorem 1.5) is not circular: the classes ω_{m,q−ζ_m} are defined from explicit q-Pochhammer and B_m normalisation data, and the proof establishes gluing via the independently computed Frobenius action on Fermat-curve forms and a base-change isomorphism. No parameter is fitted to force the claimed membership, and the q-Picard-Fuchs equation (98) is the known q-difference equation for Heine's series, not an equation inferred from the class. The main self-citation is in the secondary push-forward method, where Theorem 1.13 is restated from [17] and its proof uses [17, Eqn.(169)]; this is a minor load-bearing citation, but it does not make the whole derivation circular. A separate correctness concern, not a circularity, is that Definition 1.2 requires ordinary convergent power series in (11), while the classes in (26) are explicitly written in (q−ζ_m)^{−n−1}[[q−ζ_m]] and Theorem 2.9 only proves convergence of log(q)^{n+1} times the relevant series; as written this is an internal compatibility gap needing either a pole-cancellation proof or an amended definition. The paper also candidly labels the equivalence of H^n_naiv with Wagner–Scholze's Habiro cohomology as an expectation, so no circularity is hidden there.

Assumptions & free parameters 0 free parameters · 4 assumptions · 1 invented entities

The central construction rests on p-adic Frobenius structures from crystalline cohomology, an unpublished finite-generation input, and the expectation that the naive cohomology matches the true Habiro cohomology. No free parameters are fitted to data.

assumptions (4)
  • domain assumption Existence and compatibility of p-Frobenius endomorphisms on H^n_dR(X/B) from comparison to crystalline cohomology.
    Used throughout Definition 1.2 and all gluing proofs; cited to [2] and standard p-adic cohomology, not independently verified in the paper.
  • domain assumption Finite generation and torsion-free projectivity of H^n_dR(X/B) after localization, and injectivity into H^n_dR(X/B)⊗Q((λ)).
    Invoked in the proof of Theorem 1.5 (end of Section 2.5) as 'kindly communicated' by Scholze and Wagner; no public proof is cited.
  • ad hoc to paper The newly defined H^n_naiv(X/B) captures the essential features of Wagner's Habiro cohomology.
    The paper calls this an expectation in Section 1.3, but the title and abstract use 'Habiro cohomology' without the caveat, so the strongest interpretation depends on this unproved bridge.
  • standard math Standard q-hypergeometric identities and asymptotic expansions of q-Pochhammer symbols at roots of unity.
    Lemmas 2.1, 2.2, q-binomial theorem, and Coleman's Frobenius matrix for Fermat curves are background results used for convergence and gluing; the paper sketches proofs but relies on [7], [22], [34].
invented entities (1)
  • Naive Habiro cohomology H^n_naiv(X/B)
    purpose: Target cohomology for the explicit cycles; collections of q-series around roots of unity glued by Frobenius.
    Newly introduced in Definition 1.2. Its relation to the actual Habiro cohomology of Wagner [41] is only expected, so it has no independent falsifiable handle in this paper.

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Pith. "Pith review of Explicit classes in Habiro cohomology." pith.science (2026). https://pith.science/paper/FDPA4QJF

@misc{pith2026250519885,
  author       = {Pith},
  title        = {Pith review of: Explicit classes in Habiro cohomology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FDPA4QJF}},
  note         = {Machine review of arXiv:2505.19885}
}
abstract

We propose a cycle description of the Habiro cohomology of a smooth variety $X$ over the spectrum $B$ of an \'etale $Z[\lambda]$-algebra and construct explicit nontrivial cycles using either the Picard-Fuchs equation on $X/B$ of a hypergeometric motive, or a push-forward of elements of the Habiro ring of $X/B$. In particular, we give explicit classes for 1-parameter Calabi--Yau families. The $q$-hypergeometric origin of our cycles imply that they generate $q$-holonomic modules that define $q$-deformations of the classical Picard-Fuchs equation. We illustrate our theorems with three examples: the Legendre family of elliptic curves, the $A$-polynomial curve of the figure eight knot, and for the quintic three-fold, whose $q$-Picard Fuchs equation appeared in its genus $0$-quantum $K$-theory. Our methods give a unified treatment of quantum $K$-theory and complex Chern-Simons theory around higher dimensional critical loci.

Figures

Figures reproduced from arXiv: 2505.19885 by the authors.

Figure 1
Figure 1. The projection Y → Gm. But now, formal Gaussian integration cannot be performed on the sum (213) because the Hessian vanishes identically on X ⊂ Y . (Formal Gaussian integration can be performed at the 4 isolated points pj and p1 and p2 contribute to the asymptotics, whereas the other two do not; see [20, Eqn.(3)]). Let us explain this problem briefly. To perform formal Gaussian integration, one replaces the quantum… view at source ↗
Figure 2
Figure 2. Henkel contours on the curve X/Z[1/30] over u = log(x)-plane. The red curves cut of the principal branch of p ∆(exp(u)). When integrating this function one must change the sign of the integrand when crossing the red lines in order to perform the correct analytic continuation. Remark 4.3. The residues of ω and ω ′ at b1, . . . , b4 is zero and they vanish at ∞. We now give more coefficients of (220) expanded at q = 1… view at source ↗
Figure 3
Figure 3. The Newton polygon of the 6th order operator (251). length 2 and slope -2 are given by e0 : (q − w) 2 (q 2 − w) 2 (−1 + w) 2 e1 : 1 − q 2w − q 3w + q 7w 2 e2 : (−1 + qw) 2 e3 : q 17 − q 9w − q 10w + w 2 . (253) We did two sanity checks of the equation (251). Namely, we computed the series gE/B(q)+ O(q−1)12 (whose first two terms are given in (248)), as well as the series gE/B(q)+O(q−1)50 when λ = q ℓ for ℓ = 0, . . … view at source ↗

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Works this paper leans on

44 extracted references · 42 canonical work pages

  1. [1]

    Calabi-Yau manifolds database

    AESZ. Calabi-Yau manifolds database. Calabi-Yau manifolds database

    work page

  2. [2]

    G´ eom´ etrie rigide et cohomologie des vari´ et´ es alg´ ebriques de caract´ eristiquep

    Pierre Berthelot. G´ eom´ etrie rigide et cohomologie des vari´ et´ es alg´ ebriques de caract´ eristiquep. Num- ber 23, pages 3, 7–32. 1986. Introductions aux cohomologies p-adiques (Luminy, 1984)

    work page 1986

  3. [3]

    Prisms and prismatic cohomolo gy

    Bhargav Bhatt and Peter Scholze. Prisms and prismatic cohomolo gy. Ann. of Math. (2) , 196(3):1135– 1275, 2022

    work page 2022

  4. [4]

    p-adic gamma functions and Dwork cohomology

    Maurizio Boyarsky. p-adic gamma functions and Dwork cohomology. Trans. Amer. Math. Soc. , 257(2):359–369, 1980

    work page 1980

  5. [5]

    A p air of Calabi-Yau manifolds as an exactly soluble superconformal theory

    Philip Candelas, Xenia de la Ossa, Paul Green, and Linda Parkes. A p air of Calabi-Yau manifolds as an exactly soluble superconformal theory. Nuclear Phys. B , 359(1):21–74, 1991

    work page 1991

  6. [6]

    Local Zeta Functions From Calabi-Yau Differential Equations

    Philip Candelas and Duco van Straten. Local Zeta Functions From Calabi-Yau Differential Equations. Preprint 2021, arXiv:2104.07816

    work page Pith review arXiv 2021

  7. [7]

    On the Frobenius matrices of Fermat curves

    Robert Coleman. On the Frobenius matrices of Fermat curves. I n p-adic analysis (Trento, 1989) , volume 1454 of Lecture Notes in Math. , pages 173–193. Springer, Berlin, 1990

    work page 1989

  8. [8]

    Plane curves associated to character varieties of 3-manifolds

    Daryl Cooper, Marc Culler, Henry Gillet, Daryl Long, and Peter S halen. Plane curves associated to character varieties of 3-manifolds. Invent. Math. , 118(1):47–84, 1994

    work page 1994

Show all 44 references
  1. [9]

    Mirror symmetry and algebraic geometry , volume 68 of Mathematical Surveys and Monographs

    David Cox and Sheldon Katz. Mirror symmetry and algebraic geometry , volume 68 of Mathematical Surveys and Monographs . American Mathematical Society, Providence, RI, 1999

    1999

  2. [10]

    3-manifolds and 3d indices

    Tudor Dimofte, Davide Gaiotto, and Sergei Gukov. 3-manifolds and 3d indices. Adv. Theor. Math. Phys., 17(5):975–1076, 2013

    2013

  3. [11]

    The quantum content of the gluing equations

    Tudor Dimofte and Stavros Garoufalidis. The quantum content of the gluing equations. Geom. Topol., 17(3):1253–1315, 2013

    2013

  4. [12]

    Quantum modularity an d complex Chern-Simons theory

    Tudor Dimofte and Stavros Garoufalidis. Quantum modularity an d complex Chern-Simons theory. Com- mun. Number Theory Phys. , 12(1):1–52, 2018

    2018

  5. [13]

    1-efficient triangulations and the index of a cusped hyperbolic 3-manifold

    Stavros Garoufalidis, Craig Hodgson, Hyam Rubinstein, and Hen ry Segerman. 1-efficient triangulations and the index of a cusped hyperbolic 3-manifold. Geom. Topol., 19(5):2619–2689, 2015

    2015

  6. [14]

    A meromorphic extens ion of the 3D index

    Stavros Garoufalidis and Rinat Kashaev. A meromorphic extens ion of the 3D index. Res. Math. Sci. , 6(1):Paper No. 8, 34, 2019

    2019

  7. [15]

    Irreducibility o f q-difference operators and the knot 7 4

    Stavros Garoufalidis and Christoph Koutschan. Irreducibility o f q-difference operators and the knot 7 4. Algebr. Geom. Topol., 13(6):3261–3286, 2013

    2013

  8. [16]

    On the quantu m K-theory of the quintic

    Stavros Garoufalidis and Emanuel Scheidegger. On the quantu m K-theory of the quintic. SIGMA Sym- metry Integrability Geom. Methods Appl. , 18:Paper No. 021, 20, 2022

    2022

  9. [17]

    The Habiro ring of a number field

    Stavros Garoufalidis, Peter Scholze, Campbell Wheeler, and Do n Zagier. The Habiro ring of a number field. Preprint 2024, arXiv:2412.04241

    2024 arXiv

  10. [18]

    Perturbative invariants of cusped hy- perbolic 3-manifolds

    Stavros Garoufalidis, Matthias Storzer, and Campbell Wheeler . Perturbative invariants of cusped hy- perbolic 3-manifolds. Preprint 2023, arXiv:2305.14884. EXPLICIT CLASSES IN HABIRO COHOMOLOGY 47

    2023 arXiv

  11. [19]

    Modular q-holonomic modules

    Stavros Garoufalidis and Campbell Wheeler. Modular q-holonomic modules. Preprint 2022, arXiv:2203.17029

    2022 arXiv

  12. [20]

    Periods, the merom orphic 3D-index and the Turaev–Viro invariant

    Stavros Garoufalidis and Campbell Wheeler. Periods, the merom orphic 3D-index and the Turaev–Viro invariant. Preprint 2022, arXiv:2209.02843

    2022 arXiv

  13. [21]

    Knots, perturbative ser ies and quantum modularity

    Stavros Garoufalidis and Don Zagier. Knots, perturbative ser ies and quantum modularity. SIGMA Symmetry Integrability Geom. Methods Appl. , 20:Paper No. 055, 2024

    2024

  14. [22]

    Basic hypergeometric series , volume 96 of Encyclopedia of Math- ematics and its Applications

    George Gasper and Mizan Rahman. Basic hypergeometric series , volume 96 of Encyclopedia of Math- ematics and its Applications . Cambridge University Press, Cambridge, second edition, 2004. Wit h a foreword by Richard Askey

    2004

  15. [23]

    On the periods of certain rational integrals

    Phillip Griffiths. On the periods of certain rational integrals. I, II . Ann. of Math. (2) , 90:460–495; 90 (1969), 496–541, 1969

    1969

  16. [24]

    On the periods of abelian integrals and a formula of Chowla and Selberg

    Benedict 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

  17. [25]

    F.H. Jackson. On q-definite integrals. Quarterly Journal of Pure and Applied Mathematics , 41:193–203, 1910

    1910

  18. [26]

    p-adic cohomology: from theory to practice

    Kiran Kedlaya. p-adic cohomology: from theory to practice. In p-adic geometry , volume 45 of Univ. Lecture Ser., pages 175–203. Amer. Math. Soc., Providence, RI, 2008

    2008

  19. [27]

    p-adic differential equations , volume 125 of Cambridge Studies in Advanced Mathematics

    Kiran Kedlaya. p-adic differential equations , volume 125 of Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge, 2010

    2010

  20. [28]

    Effective p-adic cohomology for cyclic cubic threefolds

    Kiran Kedlaya. Effective p-adic cohomology for cyclic cubic threefolds. In Computational algebraic and analytic geometry , volume 572 of Contemp. Math. , pages 127–171. Amer. Math. Soc., Providence, RI, 2012

    2012

  21. [29]

    Maxim Kontsevich and Don Zagier. Periods. In Mathematics unlimited—2001 and beyond , pages 771–

    2001

  22. [30]

    Advanced Applications of the Holonomic Systems Approach

    Christoph Koutschan. Advanced Applications of the Holonomic Systems Approach . PhD thesis, RISC, Johannes Kepler University, Linz, Austria, 2009

    2009

  23. [31]

    Computing periods of rational integrals

    Pierre Lairez. Computing periods of rational integrals. Math. Comp. , 85(300):1719–1752, 2016

    2016

  24. [32]

    A p-adic analogue of the Γ-function

    Yasuo Morita. A p-adic analogue of the Γ-function. J. Fac. Sci. Univ. Tokyo Sect. IA Math. , 22(2):255– 266, 1975

    1975

  25. [33]

    Wilf, and Doron Zeilberger

    Marko Petkovˇ sek, Herbert S. Wilf, and Doron Zeilberger. A = B. A K Peters Ltd., Wellesley, MA, 1996. With a foreword by Donald E. Knuth, With a separately available compu ter disk

    1996

  26. [34]

    Hypergeometr ic motives

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

    2022

  27. [35]

    A refinement of the A-polynomial o f quivers

    Fernando Rodriguez Villegas. A refinement of the A-polynomial o f quivers. Preprint 2011, arXiv:1102.5308

    2011 arXiv

  28. [36]

    Habiro cohomology

    Peter Scholze. Habiro cohomology. Course MPIM Bonn, 2025

    2025

  29. [37]

    The Habiro ring of a number field

    Peter Scholze. The Habiro ring of a number field. Course MPIM Bonn, 2024

    2024

  30. [38]

    Canonical q-deformations in arithmetic geometry

    Peter Scholze. Canonical q-deformations in arithmetic geometry. Ann. Fac. Sci. Toulouse Math. (6) , 26(5):1163–1192, 2017

    2017

  31. [39]

    q-deformation with ( ϕ, Γ) structure of the de rham cohomology of the Legendre family of elliptic curves

    Ryotaro Shirai. q-deformation with ( ϕ, Γ) structure of the de rham cohomology of the Legendre family of elliptic curves. Preprint 2020, arXiv:2006.12310

    2020 arXiv

  32. [40]

    Multivariate Ap´ ery numbers and supercongruences of rational functions

    Armin Straub. Multivariate Ap´ ery numbers and supercongruences of rational functions. Algebra Number Theory, 8(8):1985–2007, 2014

    1985

  33. [41]

    In preparation

    Ferdinand Wagner. In preparation

  34. [42]

    Difference equation for quintic 3-fold

    Yaoxinog Wen. Difference equation for quintic 3-fold. SIGMA Symmetry Integrability Geom. Methods Appl., 18:Paper No. 043, 25, 2022

    2022

  35. [43]

    An algorithmic proof theory f or hypergeometric (ordinary and “ q”) multisum/integral identities

    Herbert Wilf and Doron Zeilberger. An algorithmic proof theory f or hypergeometric (ordinary and “ q”) multisum/integral identities. Invent. Math. , 108(3):575–633, 1992. 48 STA VROS GAROUF ALIDIS AND CAMPBELL WHEELER International Center for Mathematics, Department of Mathe m...

    1992

  36. [808]

    Springer, Berlin, 2001

    2001

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.