p-adic congruences in iterated derivatives of the Weierstrass elliptic function

Eric Peterson; Kiran Luecke

REVIEW 2 major objections 4 minor 26 references

p-adic congruences in iterated derivatives of the Weierstrass elliptic function

T0 review · 2 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Weierstrass derivatives obey p-adic Kummer congruences.

desk verdict The Witten♯ moment formulas rest on a wrong identification of y; the Todd case is fine, but Main Theorem A is unsupported as written. read the letter →

arxiv 2506.07420 v1 pith:HGNH2ZIU submitted 2025-06-09 math.NT math.AT

Kiran Luecke , Eric Peterson This is my paper · ORCID

classification math.NTmath.AT MSC 11F3311F8555N2255N3411G07

keywords p-adicmomentsequencesKummercongruencesWeierstrassellipticfunctionWittenorientationTateK-theoryE-infinityorientationscohomologysharpconstruction

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 that explicit combinations of the Weierstrass elliptic function, its iterated derivatives, and the second Eisenstein series form a p-adic moment sequence, hence satisfy the Kummer congruences attached to integer-valued polynomials on p-adic units. The argument runs a known homotopy-theoretic machine in reverse: rather than deriving E-infinity orientations from congruences, the authors start from recently established E-infinity orientations of Tate fixed-point spectra and read off the congruences those orientations force. Main Theorem A records closed formulas for the sequence terms, and Main Theorem B isolates the polar part as a separate Todd-orientation computation. The construction gives new p-adic integrality relations among elliptic functions, with explicit coefficients that a number theorist can test directly.

What carries the argument

The decisive objects are the sharp construction and the neper-moment formalism. Starting from an orientation $\omega\colon MU\to R$ with coordinate $\xi$ and formal group law $+_F$, the sharped orientation $\omega^\sharp\colon MU\to R^{tT}$ is the one with coordinate $\xi^\sharp=\xi\gamma/(\xi+_F\gamma)$, where $\gamma$ is the inverted local parameter on the Tate fixed-point spectrum $R^{tT}$. The paper computes the associated sharped exponential $\exp_{\omega^\sharp}(x)$ in terms of the Witten exponential, which is encoded by the Weierstrass $\sigma$-function, and extracts the nepers $N_n^\omega$ from $\log(x/\exp_\omega(x))$. The moments $M_n^\omega=(1-c^n)(\mathrm{id}-\psi_p/p)(N_n^\omega)$ are then p-adically integral by the moment theorem of [AHR], and unwinding the formulas gives the elliptic-function expressions of Main Theorem A. The same machinery with the Todd orientation supplies Main Theorem B.

What would settle it

Pick $p=3$ and $c=1+p=4$, take $n=1$ and $n=3$ in the Main Theorem A formulas, and test the moment identity forced by $(r^3-r)/3$: $M_3^{Wit\sharp}\equiv M_1^{Wit\sharp}\pmod 3$ in $\mathbb{Z}JqK((t))^{\wedge}_3$. Checking this against the displayed expansions to any fixed degree is finite, and one failed coefficient would disprove the theorem.

Watch

Extended reading notes

Core claim

Main Theorem A asserts that for any $c\in\mathbb{Z}_p^\times\setminus\{\pm 1\}$ there is a p-adic moment sequence valued in $\mathbb{Z}JqK((t))^{\wedge}_p$ whose terms, for $n\ge 1$, are exactly the displayed combinations of $(1-c^n)$, the difference $G_2(q)-pG_2(q^p)$ of second Eisenstein series, and differences of iterated derivatives $\wp_q^{(n-2)}$ evaluated at $\log(1-t)/(-2\pi i)$ and at $p\log(1-t)/(-2\pi i)$, with the $n=1$ and $n=2$ terms carrying extra logarithmic and Eisenstein contributions. A p-adic moment sequence is a sequence $M_n$ with the property that for every rational polynomial $f(r)=\sum_n a_n r^n$ taking $\mathbb{Z}_p^\times$ into $\mathbb{Z}_p$, the sum $\sum_n a_nM_n$ is integrally valued in the coefficient ring; this packages a family of Kummer congruences, for instance $M_p\equiv M_1\pmod p$. The proof computes the nepers of the sharped Witten orientation, applies the operation $\mathrm{id}-\psi_p/p$, and invokes the general theorem that an $\mathbb{E}_\infty$ orientation of a $K(1)$-local ring produces a p-adic moment sequence from its moments.

Load-bearing premise

Everything hinges on the assertion that the Witten orientation can be promoted to a strictly commutative, E-infinity map of spectra; the paper cites a similar case in [AHR] but gives no proof for this exact promotion, and the congruence conclusion needs it.

Editorial extensions

If this is right

For every p-adically integer-valued test polynomial $f$, the combination $\sum_n a_nM_n^{Wit\sharp}$ lies in $\mathbb{Z}JqK((t))^{\wedge}_p$; in particular $M_p^{Wit\sharp}\equiv M_1^{Wit\sharp}\pmod p$ holds.
The terms of vanishing t-degree in Main Theorem A match the measure of [Kat77, Lemma 3.5.6], and the terms of vanishing q-degree match the measure of [Lan90, Section 4.3], so the new sequence contains these classical p-adic measures as special slices.
The Todd-orientation version in Main Theorem B produces a moment sequence built from iterated finite differences, recovering the classical Bernoulli-number moment sequence as the underlying Todd case.
Because the sharp construction preserves $\mathbb{E}_\infty$ structures, the same proof gives p-adic moment sequences from any other $\mathbb{E}_\infty$ orientation, including the string orientation of tmf and Morava $E$-theories.

Reading between the lines

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

A direct p-adic proof may be possible: since the formulas are explicit, one could try to verify the Kummer congruences by q-expansion manipulations alone, which would decouple the number-theoretic conclusion from the homotopy-theoretic input.
The object behind Main Theorem A is likely a p-adic measure valued in Jacobi forms; identifying it with the construction of [Sof97] would connect the sharped elliptic genus to existing p-adic interpolation of Jacobi forms.
Applying the same moment computation to the $\sigma$-orientation of tmf would test whether the congruences persist in spectra with torsion, where $KU^{Tate}$-valued invariants cannot see everything.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The thing to know: the central calculation for Main Theorem A conflates the Todd and Witten logarithms. Definition 2.11 fixes y by exp_Wit(y) = γ, i.e. exp_Wit(y) = β^{-1} t. The proof instead sets y = -log(1-t), which solves exp_Td(y) = t. These agree only when the infinite product in exp_Wit is trivial, i.e. at q = 0. Since exp_Wit(y) = β^{-1}(1-e^{-y}) ∏_{j≥1} (1-e^{-y}q^j)(1-e^{y}q^j)/(1-q^j)^2, the substitution changes the location of the pole of ℘ and all q-dependent coefficients. So the expansion in §3.2 leading to ℘_q(log(1-t)/(-2πi)) and the moment formulas in Main Theorem A does not follow from Definition 2.11. This is a load-bearing gap, not a cosmetic one.

What the paper does well: the strategy is sensible—use CL25's theorem that sharping preserves E∞, then run the AHR machine backwards. The Todd case (Main Theorem B) is worked carefully; the neper calculation and the finite-difference formula are transparent, and I don't see an error there. The exposition is clear, and the paper is honest that it is largely an application of existing work. The congruence tables are a nice visualization.

The soft spots, in proportion: the y issue is the big one; it affects every term in Main Theorem A. Separately, Remark 2.21 asserts the E∞ structure on the Witten orientation follows by changing the target ring in AHR's Proposition 10.10; that is plausible but under-documented. The moment calculation for Main Theorem A is also omitted—"the moments are then calculated"—which is hard to verify, especially given the y problem. The reader's concern about circularity is not justified; the central claim is not circular.

Who this is for: number theorists interested in p-adic measures and elliptic functions, and homotopy theorists working on complex orientations. The Todd part is a nice explicit computation, but the advertised novelty is the Witten part, and it is currently broken.

Recommendation: send it to a serious referee. The mistake is concrete and fixable in principle—one would need to redo the expansion with the true Witten-logarithm y(t), and the formulas would likely change. If corrected, it could be a solid contribution. As it stands, I would not rely on Main Theorem A.

Referee Report

2 major / 4 minor

Summary. The paper claims two p-adic moment-sequence theorems. Main Theorem B gives explicit moments for the sharped Todd orientation in terms of finite differences and the element t, while Main Theorem A gives explicit moments for the sharped Witten orientation, expressed through differences of the second Eisenstein series and iterated derivatives of the Weierstrass elliptic function evaluated at q and q^p. The proofs combine the sharp construction of Ando–French–Ganter, the recent E-infinity orientation theorem of Carmeli–Luecke, and the Ando–Hopkins–Rezk theorem that E-infinity orientations of K(1)-local rings produce p-adic moment sequences. The paper is written as a short application note, with the main computational work in Sections 3.1 and 3.2.

Significance. If the two main theorems are correct, this is an attractive new application of E-infinity orientation theory to concrete p-adic congruences, and the explicit formulas give elliptic-function witnesses to Kummer congruences that are not obviously accessible by elementary means. The paper is clearly organized, and the computational parts for the Todd sharp orientation in Section 3.1 are convincing. The paper also profits from stating precise moment formulas rather than only existence statements, and the accompanying tables provide useful empirical confirmation. However, the proof of the main Witten sharp calculation currently rests on two load-bearing inputs that are either identified incorrectly or asserted without proof: the coordinate identification in Section 3.2 and the E-infinity structure on the Witten orientation in Remark 2.21. Both must be addressed before the central claim can be regarded as established.

major comments (2)

[§3.2, Definition 2.11] At the beginning of Section 3 the authors set 2πiα = y = −log(1−t), i.e. t = 1−e^{−y}. Definition 2.11, however, defines the sharped exponential from an element y satisfying exp_ω(y) = γ. For ω = ω_Wit, after the degree-zero normalization t = βγ this condition reads t = (1−e^{−y})∏_{j≥1}(1−e^{−y}q^j)(1−e^{y}q^j)/(1−q^j)^2, not t = 1−e^{−y}. The expansion in the proof of Main Theorem A of log(exp_Wit(x+y)/exp_Wit(y)) in terms of ℘_q(log(1−t)/(−2πi)) is therefore not the logarithm associated to the sharped Witten orientation as defined, unless the infinite product is 1 in ZJqK((t))^∧_p. The authors need either to use the y that actually solves exp_Wit(y) = γ and recompute the moments, or to prove that the product factor is trivial in the relevant completed ring. Because the explicit formulas in Main Theorem A are the central claim, this gap is load-bearing.
[Remark 2.21] Remark 2.21 asserts that ω_Wit: MU → (KU^Tate)^∧_p admits an E-infinity structure by changing the target in [AHR, Proposition 10.10] or by obstruction theory, but no proof or reference is supplied. Theorem 2.18, which is the source of the moment-sequence property, requires an E-infinity orientation. The cited Proposition 10.10 is stated for an MU⟨6⟩-orientation, and the passage from MU⟨6⟩ to MU together with the target replacement needs argument. Please supply a complete proof or a precise citation that covers the Witten orientation itself.

minor comments (4)

[Definition 2.6] The text says 'cf. Theorem 2.4' but the intended reference is Example 2.4, where the additive orientation is discussed.
[§3.1 and §3.2] Several cross-references are to the wrong numbered item: 'Theorem 2.7' should be 'Example 2.7', 'Theorem 2.11' should be 'Definition 2.11', and in the proof of Main Theorem A 'Theorem 2.2' should be 'Example 2.2' and 'Theorem 3.3' should be 'Lemma 3.3'.
[Figure 4] The caption of Figure 4 repeats the sentence 'Congruences here relate the colored digits across different tables rather than within columns.'
[Title figure and figures] The large table of binary expansions at the top of the paper is not referenced in the text, and the figure captions refer to colors that are not described for grayscale printing; please add a pointer and a color description.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the target moment formulas are computed from nepter series via a fixed linear operator, not assumed as inputs; the cited E-infinity inputs are external general theorems.

full rationale

The derivation chain is not circular. Main Theorem A obtains each M_n^{Wit♯} as (1−c^n)(id − ψ_p/p)(N_n^{Wit♯}), where the nepers N_n^{Wit♯} are computed in Section 3.2 from the sharped Witten exponential formula log(x/exp_Wit(x)) + log(exp_Wit(x+y)/exp_Wit(y)). No quantity appearing in the conclusion is fed back as an input: the explicit ℘-derivative and Eisenstein-series expressions are outputs of expanding known generating functions, and the p-adic moment-sequence property is imported from the external Theorem 2.18 (AHR Theorem 6.1). The CL25 theorem that the sharp construction preserves E∞ orientations is a general homotopy-theoretic result whose assumptions do not include the target congruences, so even though it involves the first author, it is not a restatement or fitted version of the claimed moments. The possible correctness gaps flagged in the manuscript — Remark 2.21's unproved extension of AHR Proposition 10.10 to the full Witten orientation, and the Section 3.2 identification y = −log(1−t) versus Definition 2.11's defining condition exp_Wit(y) = γ — are mathematical-support or computation issues, not circularity: even if they invalidate the displayed formulas, the conclusion is not definitionally equal to the inputs. Hence a score of 0 is appropriate for circularity.

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

The paper introduces no new free parameters or entities. The moments are computed, not fitted. The central claim rests on a chain of existing theorems plus one unproven assertion about the Witten orientation.

assumptions (4)

domain assumption The sharp construction preserves E-infinity orientations (Theorem 2.13, from [CL25, Theorem 5.14]).
This is the main external input from the first author's recent work; it is used to conclude that the sharped Todd and Witten orientations are E-infinity.
domain assumption An E-infinity orientation of a K(1)-local ring flat over KU_p^ produces a p-adic moment sequence (Theorem 2.18, from [AHR, Theorem 6.1]).
This is the bridge from topology to number theory; it converts the E-infinity structure into integrality congruences.
ad hoc to paper The Witten orientation ω_Wit: MU → (KU^Tate)^_p admits an E-infinity structure (Remark 2.21).
The paper asserts this without proof, as an extension of [AHR, Proposition 10.10]. Main Theorem A requires it.
domain assumption The target rings (KU^{tT})^_p and (KU^{tT,Tate})^_p are flat over KU_p^ and K(1)-local.
Needed to apply Theorem 2.18; the paper states this in Section 2.3 without detailed verification.

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Cite this review

Pith. "Pith review of p-adic congruences in iterated derivatives of the Weierstrass elliptic function." pith.science (2026). https://pith.science/paper/HGNH2ZIU

@misc{pith2026250607420,
  author       = {Pith},
  title        = {Pith review of: p-adic congruences in iterated derivatives of the Weierstrass elliptic function},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HGNH2ZIU}},
  note         = {Machine review of arXiv:2506.07420}
}

abstract

We use homotopy theoretic methods to prove congruence relations of number theoretic interest. Specifically, we use the theory of $\mathbb E_\infty$ complex orientations to establish $p$-adic K\"ummer congruences among iterated derivatives of the Weierstrass elliptic function. The machinery of Ando, Hopkins, and Rezk was developed with the intended application of taking congruence relations as input and producing $\mathbb E_\infty$-orientations as output. We run their machine in reverse, using as input the recent results of Carmeli and the first author on the existence of $\mathbb E_\infty$-orientations of Tate fixed-point objects.

Figures

Figures reproduced from arXiv: 2506.07420 by the authors.

**Figure 2.** Some 3–adic Witten♯ moments at c = 1 + p, in a range of t– and q–degrees. Congruences here relate the colored digits across different tables rather than within columns. The relevant test polynomials are p −2 (r (p−1)p − 1) and p −3 (r (p−1)p 2 − 1). 17 [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

**Figure 3.** A range of binary expansions of the 2–adic Todd i [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

Figures from the paper (1 more)

**Figure 4.** Figure 4: Some 2–adic Witten♯ moments at c = 1 + p, in a range of t– and q–degrees. Congruences here relate the colored digits across different tables rather than within columns. Congruences here relate the colored digits across different tables rather than within columns. The r…

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Pith tools

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

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J. F. Adams. Stable homotopy and generalised homology . Chicago Lectures in Mathematics. University of Chicago Press, Chicago, Ill.-London, 1974

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Matthew Ando, Christopher P. French, and Nora Ganter. The J acobi orientation and the two-variable elliptic genus. Algebr. Geom. Topol. , 8(1):493--539, 2008

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R. Burklund, T. Schlank, and A. Yuan. The chromatic nullstellensatz. Ann. Math , 2024

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Shachar Carmeli and Kiran Luecke. Tate-valued characteristic classes, 2025, 2503.12134

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