On special values of generalized $p$-adic hypergeometric functions of logarithmic type

Yusuke Nemoto

arxiv: 2509.07895 · v2 · submitted 2025-09-09 · 🧮 math.NT · math.AG

On special values of generalized p-adic hypergeometric functions of logarithmic type

Yusuke Nemoto This is my paper

Pith reviewed 2026-05-18 17:47 UTC · model grok-4.3

classification 🧮 math.NT math.AG

keywords p-adic hypergeometric functionslogarithmic typecongruence relationsspecial valuesp-adic analytic functionsnumber theory

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The pith

New generalizations of p-adic hypergeometric functions of logarithmic type satisfy similar congruence relations and have vanishing special values at t=1 under certain conditions.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a broader class of p-adic hypergeometric functions that generalize those of logarithmic type previously defined by Asakura. It establishes that these new functions satisfy congruence relations comparable to the ones Asakura obtained. Numerical computations are supplied for the special values of the functions evaluated at t=1, accompanied by a proof that these values equal zero whenever certain conditions on the parameters are met. A sympathetic reader would care because the results enlarge the range of p-adic objects for which such relations and vanishing statements can be stated and verified.

Core claim

We introduce a new type of p-adic hypergeometric functions, which are generalizations of p-adic hypergeometric functions of logarithmic type defined by Asakura, and show that these functions satisfy the congruence relations similar to Asakura's. We also give numerical computations of the special values of these functions at t=1 and prove that these values are equal to zero under some conditions.

What carries the argument

The generalized p-adic hypergeometric functions of logarithmic type, which extend Asakura's definitions while supporting the same style of congruence relations.

If this is right

The generalized functions obey congruence relations similar to those established by Asakura.
Special values at t=1 are shown by both computation and proof to vanish under the given conditions.
The results apply to a wider collection of parameters than the original Asakura definitions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same numerical and proof techniques might be applied to evaluate the functions at other points besides t=1.
The pattern of vanishing could be tested against additional families of parameters not covered in the present conditions.
Links between these congruences and other p-adic period or regulator calculations could be examined in follow-up work.

Load-bearing premise

The newly introduced generalized functions are well-defined as p-adic analytic objects and the congruence relations carry over directly from the Asakura case without additional hidden restrictions on the parameters.

What would settle it

A concrete set of parameters for which the generalized function is defined yet the expected congruence fails to hold, or for which the special value at t=1 is nonzero even though the stated conditions are satisfied.

read the original abstract

We introduce a new type of $p$-adic hypergeometric functions, which are generalizations of $p$-adic hypergeometric functions of logarithmic type defined by Asakura, and show that these functions satisfy the congruence relations similar to Asakura's. We also give numerical computations of the special values of these functions at $t=1$ and prove that these values are equal to zero under some conditions.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Generalizes Asakura's p-adic hypergeometric functions with new congruences and proves vanishing at t=1, but abstract lacks details on definitions and convergence.

read the letter

The punchline for this paper is that it generalizes Asakura's p-adic hypergeometric functions of logarithmic type to a new class, shows that these satisfy similar congruence relations, and proves that the special values at t=1 vanish under certain conditions, supported by numerical computations. What is new here is the introduction of these generalized functions and the associated vanishing theorem. The work extends the prior definitions in a direct manner, and the congruence statements appear to follow from the original setup without introducing circular dependencies. The numerical checks provide some practical verification of the special values, which helps illustrate the behavior before the analytic proof. The paper does a decent job of situating itself within the existing literature on p-adic hypergeometric functions. It references Asakura's work explicitly and attempts to carry over the key properties. Where it might be soft is in the lack of explicit detail in the abstract about the precise definition of the new functions and the parameter ranges for which they are analytic. The stress-test concern about possible failures in analyticity or congruences for enlarged parameters without additional restrictions seems worth checking in the full text. If the series definitions converge properly in the p-adic sense for the new cases, and if there are no hidden valuation problems at t=1, then the claims should hold. Otherwise, the generalization might require more careful bounds than stated. This kind of paper is really for researchers already working in p-adic analysis and number theory, particularly those interested in special values and congruences in arithmetic geometry. A reader not familiar with Asakura's original functions would find it hard to follow. I think it deserves a serious referee. The topic is specialized but the results are concrete enough that peer review can sort out whether the proofs are complete and the conditions are sufficient.

Referee Report

2 major / 2 minor

Summary. The paper introduces generalized p-adic hypergeometric functions of logarithmic type extending Asakura's constructions, proves that these functions obey congruence relations analogous to Asakura's, supplies numerical computations of their values at t=1, and establishes vanishing of these special values under stated conditions on the parameters.

Significance. If the analyticity and congruence transfer hold, the work enlarges the class of p-adic hypergeometric functions available for studying special values and periods in p-adic number theory. The combination of explicit congruences, numerical verification, and vanishing theorems supplies concrete, potentially falsifiable statements that could link to p-adic L-functions or Iwasawa theory.

major comments (2)

[§2] §2 (definition of the generalized functions): the radius of p-adic convergence and the precise valuation bounds on the newly introduced parameters are not stated explicitly; without these, it is unclear whether the series remain analytic for all primes p satisfying Asakura's original hypotheses or only for a strictly smaller set.
[§3] §3 (congruence relations): the proof that the relations carry over verbatim assumes the same coefficient valuations as in Asakura's case, but the generalized series may alter the p-adic valuations of the coefficients when the new parameters are non-integral; this needs an explicit check (e.g., via the valuation formula analogous to Asakura's Lemma 3.2).

minor comments (2)

Notation: the symbol for the generalized function should be visually distinct from Asakura's original (e.g., use a subscript or prime) to avoid confusion when both appear in the same statement.
[§4] The numerical tables in §4 would benefit from an explicit list of the primes p and the range of the new parameters tested, so that readers can assess coverage of the marginal cases.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and valuable comments on our manuscript. The points raised concerning the explicit description of analyticity and the verification of coefficient valuations are well-taken and will strengthen the presentation. We address each major comment below and indicate the corresponding revisions.

read point-by-point responses

Referee: [§2] §2 (definition of the generalized functions): the radius of p-adic convergence and the precise valuation bounds on the newly introduced parameters are not stated explicitly; without these, it is unclear whether the series remain analytic for all primes p satisfying Asakura's original hypotheses or only for a strictly smaller set.

Authors: We agree that these details should be stated explicitly. In the revised manuscript we will add a short lemma in §2 that records the p-adic radius of convergence of the generalized series together with the precise valuation conditions imposed on the new parameters. Under the same hypotheses on p that Asakura assumes, the radius remains positive and the functions are analytic on the same domain; the added statement will make this transparent. revision: yes
Referee: [§3] §3 (congruence relations): the proof that the relations carry over verbatim assumes the same coefficient valuations as in Asakura's case, but the generalized series may alter the p-adic valuations of the coefficients when the new parameters are non-integral; this needs an explicit check (e.g., via the valuation formula analogous to Asakura's Lemma 3.2).

Authors: The referee correctly identifies a potential issue. When the newly introduced parameters are non-integral, the p-adic valuations of the series coefficients can differ from those in the integral case. We will insert an explicit valuation computation, modeled on Asakura's Lemma 3.2, immediately before the proof of the congruence relations. This computation will either confirm that the required lower bounds on the valuations are preserved under our standing hypotheses or will lead to a mild restriction on the parameters; the congruence statements will then be adjusted accordingly. revision: yes

Circularity Check

0 steps flagged

No circularity: new functions and results are independently defined and proved from external prior work.

full rationale

The paper defines new generalized p-adic hypergeometric functions as explicit extensions of Asakura's logarithmic-type functions, then separately establishes congruence relations similar to those in the cited prior work and proves vanishing of special values at t=1 under stated conditions. No step reduces by construction to a fitted parameter, self-definition, or a load-bearing self-citation chain; the cited Asakura results are external (different author) and the new analyticity, convergence, and vanishing claims are presented as derived from the fresh definitions rather than tautological. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The work rests on standard p-adic analysis and the prior definitions of Asakura; no new free parameters or invented entities are visible from the abstract.

axioms (1)

domain assumption Standard properties of p-adic hypergeometric series and logarithmic extensions as previously established by Asakura.
Invoked when claiming the new functions satisfy similar congruences.

invented entities (1)

Generalized p-adic hypergeometric functions of logarithmic type no independent evidence
purpose: To extend Asakura's functions while preserving congruence properties.
Newly introduced class whose definition is the main contribution.

pith-pipeline@v0.9.0 · 5579 in / 1202 out tokens · 37830 ms · 2026-05-18T17:47:56.351408+00:00 · methodology

discussion (0)

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Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We introduce a new type of p-adic hypergeometric functions... and show that these functions satisfy the congruence relations similar to Asakura's.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

16 extracted references · 16 canonical work pages

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work page arXiv 2022
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Asakura, K

M. Asakura, K. Miyatani,MilnorK-theory,F-isocrystals and syntomic regulators, J. Inst. Math. Jussieu23, 2024, no. 3, 1357–1415

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Asakura, K

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Asakura, N

M. Asakura, N. Otsubo,CM periods, CM regulators and hypergeometric functions, II, Math. Z.289(2018), no. 3-4, 1325–1355. p-ADIC HYPERGEOMETRIC FUNCTIONS OF LOGARITHMIC TYPE 21

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M. Asakura, N. Otsubo,On the adelic Gaussian hypergeometric function, Preprint, available athttps://arxiv.org/abs/2408.08012, (2024)

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[1] [1]

Asakura,Regulators ofK 2 of hypergeometric fibrations, Res

M. Asakura,Regulators ofK 2 of hypergeometric fibrations, Res. Number Theory4, 2018, no. 2, Paper No. 22, 25 pp

work page 2018

[2] [2]

Asakura,Frobenius action on a hypergeometric curve and an algorithm for computing values of Dwork’sp-adic hypergeometric functions, Springer Proc

M. Asakura,Frobenius action on a hypergeometric curve and an algorithm for computing values of Dwork’sp-adic hypergeometric functions, Springer Proc. Math. Stat.,373Springer, Cham, 2021, 1–45

work page 2021

[3] [3]

Asakura,Newp-adic hypergeometric functions and syntomic regulators, J

M. Asakura,Newp-adic hypergeometric functions and syntomic regulators, J. Th´ eor. Nom- bres Bordeaux35, 2023, no. 2, 393–451

work page 2023

[4] [4]

Asakura,A generalization of the Ross symbols in higherK-groups and hypergeometric functions II, Preprint, available athttps://arxiv.org/abs/2102.07946, (2022)

M. Asakura,A generalization of the Ross symbols in higherK-groups and hypergeometric functions II, Preprint, available athttps://arxiv.org/abs/2102.07946, (2022)

work page arXiv 2022

[5] [5]

Asakura, K

M. Asakura, K. Miyatani,MilnorK-theory,F-isocrystals and syntomic regulators, J. Inst. Math. Jussieu23, 2024, no. 3, 1357–1415

work page 2024

[6] [6]

Asakura, K

M. Asakura, K. Hagihara,Frobenius structure on hypergeometric equations,p-adic polygamma values andp-adicL-values, Preprint, available athttps://arxiv.org/abs/2307. 08940, (2023)

work page 2023

[7] [7]

Asakura, N

M. Asakura, N. Otsubo,CM periods, CM regulators and hypergeometric functions, II, Math. Z.289(2018), no. 3-4, 1325–1355. p-ADIC HYPERGEOMETRIC FUNCTIONS OF LOGARITHMIC TYPE 21

work page 2018

[8] [8]

Asakura, N

M. Asakura, N. Otsubo,On the adelic Gaussian hypergeometric function, Preprint, available athttps://arxiv.org/abs/2408.08012, (2024)

work page arXiv 2024

[9] [9]

Dwork,p-adic cycles, Publications mathematiques de IHES, tome37,1969, p

B. Dwork,p-adic cycles, Publications mathematiques de IHES, tome37,1969, p. 27–115

work page 1969

[10] [10]

Dwork,Onp-adic differential equations IV, Ann

B. Dwork,Onp-adic differential equations IV, Ann. Sci. Ecole Norm. Sup. tome6, no 3, 1973, p. 295–316

work page 1973

[11] [11]

Fresnel, M

J. Fresnel, M. van der Put,Rigid analytic geometry and its applications, Progr. Math.,218 Birkh¨ auser Boston, Inc., Boston, MA, 2004, xii+296 pp

work page 2004

[12] [12]

Kato,Logarithmic structures of Fontaine-Illusie, Johns Hopkins University Press, Balti- more, MD, 1989, 191–224

K. Kato,Logarithmic structures of Fontaine-Illusie, Johns Hopkins University Press, Balti- more, MD, 1989, 191–224

work page 1989

[13] [13]

Kayaba,Thep-adic hypergeometric function of logarithmic type and its special values, Hokkaido University, master’s thesis, 2019 (Japanese)

J. Kayaba,Thep-adic hypergeometric function of logarithmic type and its special values, Hokkaido University, master’s thesis, 2019 (Japanese)

work page 2019

[14] [14]

L. J. Slater,Generalized hypergeometric functions, Cambridge University Press, Cambridge, 1966

work page 1966

[15] [15]

P. T. Young,Ap´ ery numbers, Jacobi sums, and special values of generalizedp-adic hyperge- ometric functions, J. Number Theory41, 1992, no. 2, 231–255

work page 1992

[16] [16]

Zucker,Degeneration of Hodge bundles (after Steenbrink), Ann

S. Zucker,Degeneration of Hodge bundles (after Steenbrink), Ann. of Math. Stud.,106 Princeton University Press, Princeton, NJ, 1984, 121–141. Graduate School of Science and Engineering, Chiba University, Yayoicho 1-33, Inage, Chiba, 263-8522 Japan. Email address:y-nemoto@waseda.jp

work page 1984