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REVIEW 3 major objections 3 minor 34 references

Cusp forms as $p$-adic limits circumventing $p$-adic version of the Legendre period relation

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Pith's one-line read The paper proves the p-adic constant γ is nonzero for all but finitely many supersingular primes, closing a gap in earlier constructions of cusp forms as p-adic limits.

desk verdict A genuinely new infinite-family non-vanishing result in p-adic mock modular form coupling, but the proof as written has sign errors and an unjustified strict isomorphism claim that need repair. read the letter →

arxiv 2506.07107 v2 pith:WO52OUCL submitted 2025-06-08 math.NT

Pavel Guerzhoy This is my paper
classification math.NT MSC 11F1111F3314H52
keywords cuspformsp-adiclimitsmockmodularformalgrouplawssupersingularprimesGammafunctionellipticcurves
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 closes a gap in the program that expresses a cusp form as a $p$-adic limit of derivatives of harmonic Maass forms under repeated application of Atkin's $U$-operator. Earlier results in this direction required a certain constant $\gamma$ to be nonzero, and non-vanishing was known only in finitely many cases. The author proves that for a weight-2 newform with integer coefficients, $\gamma \neq 0$ for all but possibly finitely many supersingular primes $p$ with $b(p) = 0$, yielding an infinite family of examples. For the original example—the newform associated with the elliptic curve $y^2 = x^3 - x$—he computes $\gamma$ explicitly as a ratio of values of Morita's $p$-adic Gamma function (the $p$-adic analog of the classical Gamma function), making the non-vanishing transparent.

What carries the argument

The load-bearing object is the formal group law attached to the elliptic curve coming from the modular parametrization, together with its invariant differential. The quantity $\mu_p$ is defined by the condition that $\zeta(\Lambda, E_g) - \lambda_p E_g - \mu_p \frac{1}{p} E_g|V$ has a $p$-adically integral $q$-expansion, in direct analogy with the classical splitting of the Weierstrass zeta function. Honda's theorem gives a strict isomorphism over $\mathbb{Z}_p$ between this formal group and the one with logarithm $E_g$, and the paper introduces the $p$-typical formal group with logarithm $l_t(u) = \sum_{n \geq 0} (-1)^n \frac{u^{p^{2n}}}{p^n}$ to isolate the coefficient of $u^p$. That coefficient, multiplied by $p$, is congruent to $-\frac{1}{12} E_{p+1} \pmod{p}$; the second congruence is proved with Lang's factorization of Eisenstein polynomials into irreducibles over the algebraic closure of $\mathbb{F}_p$, showing $E_{p+1}$ cannot vanish modulo $p$ at a supersingular prime.

What would settle it

Take an elliptic curve with a supersingular prime of good reduction, for instance the curve $y^2 = 4x^3 + 16x$ at $p = 3$, and compute $\mu_p$ directly from the condition that $\zeta(\Lambda, E_g) - \lambda_p E_g - \mu_p \frac{1}{p} E_g|V$ has a $p$-adically integral $q$-expansion; if $\mu_p = 0$ for any such prime, Theorem 3 is false.

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Extended reading notes

Core claim

The central discovery is Theorem 1: under the setup described above, the exceptional $p$-adic constant $\gamma$ is nonzero for all but finitely many primes $p \nmid N$ with $b(p) = 0$. The proof passes through Theorem 3, which characterizes the $p$-adic quantity $\mu_p$ attached to the elliptic curve associated to $g$ via Eichler–Shimura theory: $\mu_p = 0$ if and only if the reduction at $p$ is ordinary. Since $b(p) = 0$ is exactly the supersingular case, this yields $\mu_p \neq 0$ and hence $\gamma \neq 0$. The argument reduces Theorem 3 to two congruences: $\mu_p \equiv -\frac{1}{12} E_{p+1} \pmod{p}$ and $E_{p+1} \not\equiv 0 \pmod{p}$ for primes of good supersingular reduction, proved using formal group laws and Lang's theory of Eisenstein polynomials. In the $N = 32$ example the paper further derives the explicit formula $\gamma = 8 \left(\frac{2}{p}\right) \frac{\Gamma_p(1/2)}{\Gamma_p(1/4)^2}$.

Load-bearing premise

The proof assumes, without proof, that the formal group law attached to the elliptic curve is strictly isomorphic over the $p$-adic integers to a particular standard height-2 formal group whose logarithm is the series $\sum_{n \geq 0} (-1)^n u^{p^{2n}}/p^n$.

Editorial extensions

If this is right

  • For any weight-2 newform with integer coefficients and infinitely many primes $p$ with $b(p)=0$, the $p$-adic limit in Proposition 1 converges to $g$ without any exceptional zero; the previously suspected obstruction $\gamma = 0$ occurs at most finitely often.
  • The non-vanishing is independent of one-dimensionality, complex multiplication, and eta-quotient presentations, so the result applies to an infinite family of cases rather than the finitely many covered by earlier methods.
  • The congruence $\mu_p \equiv -\frac{1}{12} E_{p+1} \pmod{p}$ ties the mock-modular coupling constant to the Eisenstein series $E_{p+1}$, making the relation between these $p$-adic limits and supersingular reduction explicit.
  • In the $N=32$ example, the explicit formula $\gamma = 8 \left(\frac{2}{p}\right) \frac{\Gamma_p(1/2)}{\Gamma_p(1/4)^2}$ gives a manifestly nonzero value for every $p \equiv 3 \pmod{4}$.

Reading between the lines

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

  • The proof only uses that the reduced formal group has height 2, not the finer structure of its isomorphism class; this suggests the same non-vanishing should hold for supersingular elliptic curves over number fields.
  • The explicit example produces $\mu_p^2$ where the classical complex period gives $\mu_\infty$ itself; this quadratic discrepancy hints at a deeper p-adic–complex period relation not explained in the paper, perhaps accessible through p-adic Chowla–Selberg theory.
  • A testable extension is to compute $\gamma$ via the congruence (11) for non-CM weight-2 newforms at supersingular primes; if any such computation produced $\gamma = 0$, the finite exception set in Theorem 1 would have to be made explicit.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 3 minor

Summary. The paper studies the non-vanishing of the p-adic constant γ that arises in recent results expressing a cusp form as a p-adic limit of weakly holomorphic modular forms under repeated Atkin U-operators. The author proposes a new approach: for a weight-2 newform g with b(p)=0, the exceptional constant γ is identified with a formal-group invariant µ_p of the associated elliptic curve, and the non-vanishing is reduced to a congruence µ_p ≡ -E_{p+1}/12 (mod p) together with a classical non-vanishing of E_{p+1} at supersingular primes. This yields Theorem 1, an infinite family of primes for which γ≠0, with an essentially uniform proof. The original example of El-Guindy and Ono (g=η^2(4τ)η^2(8τ), N=32) is treated in detail, and γ is computed explicitly in terms of Morita's p-adic Gamma function (Theorem 2). The proof combines Honda's formal group theorem, Dieudonné module invariants from the author's earlier work [13,14], and a coefficient-level calculation involving the Eisenstein series E_{p+1}.

Significance. If the main results are correct, the paper makes a genuine advance: it replaces the finitely many, CM-specific non-vanishing arguments of Ahlgren--Samart, Hanson--Jameson, and Dicks with a method that proves γ≠0 for all but finitely many supersingular primes in a general weight-2 setting, and it gives an explicit p-adic Gamma formula for the original N=32 example. The use of the Eisenstein series E_{p+1} as the source of non-vanishing is an appealing and potentially reusable idea. The paper also exhibits a clean connection between the mock-modular constant γ and Dieudonné-module invariants of the associated elliptic curve. However, several load-bearing points in the written proof are not correct as stated: an unjustified (and likely false) strict isomorphism claim in Section 4, a sign inconsistency between Theorem 2 and its proof, and an arithmetic error in the p=3 case. These issues do not destroy the plausibility of the main theorem, but they require substantial repair.

major comments (3)
  1. [Section 4, paragraph beginning "In order to prove Theorem 3"] The assertion that the formal group laws with logarithms Eg and l_t(u)=Σ(-1)^n u^{p^{2n}}/p^n are strictly isomorphic over Z_p whenever p is a prime of good reduction and b(p)=0 is not justified by the cited [16, Theorem 15.2.9]. Hazewinkel's theorem guarantees that every formal group over Z_p is strictly isomorphic to some p-typical formal group, but it does not identify the unit parameter of that p-typical group. For height-2 formal groups over Z_p, the strict isomorphism class is encoded in a unit modulo 1+pZ_p, and l_t corresponds to the parameter -1; a generic supersingular elliptic curve need not have this invariant. Since the subsequent computation of µ_p via ζ(Λ,l_t(u)) depends on this isomorphism, the proof of Theorem 3 has a real gap at this point. The gap is likely repairable: for any height-2 p-typical logarithm L(u)=u+O(u^p), the coefficient of u^p in ζ(Λ,L(u)) is still -2G_{p+1}/p! because 1/L has no u^p term and L(u)^p has u^p-coefficient 1, independent of the unit parameter. The paper should be revised to make this argument instead of claiming a specific strict isomorphism to l_t.
  2. [Theorem 2 and its proof in Section 3] There is a sign inconsistency between the theorem statement and the proof. Theorem 2 states γ = 8(2/p)Γ_p(1/2)/Γ_p(1/4)^2, while the proof concludes µ_p = -8(2/p)Γ_p(1/2)/Γ_p(1/4)^2. Since the author explicitly identifies γ with µ_p (with C=1 in this example), one of these formulas must be wrong. This is not merely a local typo: the minus sign appears already in the displayed limit expression for µ_p and propagates through the p-adic Gamma manipulations. The authors should correct either the theorem statement or the proof, and verify which sign is consistent with the definition (9) and with the p=3 calculation.
  3. [Section 4, p=3 calculation] The congruence -g2/20 ≡ -g2 (mod 3) is arithmetically false: since 20 ≡ 2 (mod 3), one has 1/20 ≡ -1 (mod 3), hence -g2/20 ≡ g2 (mod 3), not -g2. Consequently the displayed derivation of (11) for p=3 is incorrect. In the N=32 example with g2=-16, the computation gives µ_3 ≡ 2 (mod 3) whereas (11) would require µ_3 ≡ -g2 ≡ 1 (mod 3). The non-vanishing conclusion µ_3 ≠ 0 still follows from g2^3 ≡ Δ ≠ 0 (mod 3), so the overall theorem may survive, but (11) as stated does not hold for p=3 in this example and the p=3 case needs a corrected treatment, possibly by changing the sign in (11) or by handling p=3 separately without invoking the Kummer congruence used for p>3.
minor comments (3)
  1. [Abstract and title] The text contains several spacing artifacts in the title and abstract (e.g., "NON-V ANISHING", "CER T AIN") that should be cleaned up in the final version.
  2. [Section 4, paragraph after (11)] The phrase "µ_p is congruent modulo p to p times the coefficient of u^p" is potentially confusing. Since the coefficient of u^p in ζ(Λ,l_t(u)) is -2G_{p+1}/p!, the intended statement is that p times this coefficient is congruent to µ_p modulo p; the wording should be made unambiguous.
  3. [Remark 1] The congruence condition m ≡ (M-1)/2 (mod 2) is not explained and appears to be a provisional guess; it would help to state explicitly that this is a heuristic remark and not a proved formula.

Circularity Check

0 steps flagged · score 2.0 of 10

No circular derivation: the non-vanishing result rests on an independent Eisenstein-series congruence and on classical modular-form facts; only minor reliance on the author's earlier technical lemmas.

full rationale

The central claim is not forced by construction. Existence of beta and gamma is imported from [13, Prop 5], which is an existence statement and does not assert gamma != 0; the new content is to prove gamma != 0. The proof reduces Theorem 3 to (11), mu_p ≡ -E_{p+1}/12 mod p, and (12), E_{p+1} not ≡ 0 mod p at supersingular primes. Congruence (11) is obtained by extracting the coefficient of u^p in zeta(Lambda, l_t(u)) and using Kummer congruences; congruence (12) is proved from the standard fact that the reductions of E_{p-1} and E_{p+1} are relatively prime in the ring of modular forms mod p (Lang, Thm X.7.3(i)), so E_{p+1} is a unit exactly where the Hasse invariant E_{p-1} vanishes. Neither step fits a parameter to the desired conclusion, and the non-vanishing is an externally checkable congruence modulo p. The author's previous work [13] supplies existence of beta,gamma and [14, Cor. 1(a)] supplies invariance of lambda_p,mu_p under strict FGL isomorphisms; these are parameter-free lemmas that do not assume the non-vanishing, so they are self-citations but not circular. A separate correctness gap is the unproved assertion in Section 4 that E_g is strictly isomorphic over Z_p to the fixed p-typical logarithm l_t(u) = Σ(-1)^n u^{p^{2n}}/p^n; Hazewinkel's theorem gives existence of some p-typical FGL, not strict isomorphism to this particular one, and over Z_p height-2 p-typical FGLs are not all strictly isomorphic. This is a mathematical gap in the written proof, not a circularity; the coefficient-of-u^p calculation may be recoverable by a direct height-2 argument. Overall, no fitted-input or definitional circularity is present, so the score is 2 only for the moderate reliance on the author's earlier technical results.

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

No new entities are postulated. The central claim rests on several prior theorems, two of which are self-citations ([13], [14]); these provide the existence and the FGL-invariance used to identify constants, but the non-vanishing is not assumed from them. Free parameters: none in the final statements; α, β, λp, µp are determined rather than fitted. The main external benchmarks are the Eisenstein series E_{p+1}.

assumptions (7)
  • domain assumption Existence of a harmonic Maass form f good for g and of α ∈ C such that Fα has rational coefficients.
    Invoked in Section 1 as [7, Proposition 5.1] and [12, Theorem 1.1].
  • domain assumption [13, Proposition 5] gives unique β, γ ∈ Qp such that ord_p(Fα,β,γ) > -∞.
    Used to define γ and in the proof of Proposition 1; the present paper provides no independent proof.
  • standard math Honda's theorem: the formal group of an elliptic curve is strictly isomorphic over Z_p to a Honda formal group of matching height.
    Used in Section 4 to replace the FGL of E with a p-typical FGL and to identify Eg with the FGL logarithm.
  • domain assumption [14, Corollary 1(a)] invariance of λp, µp under strict FGL isomorphism.
    Self-citation; needed to identify the computed constants with the original γ.
  • standard math Dieudonné module D(G) is free of rank 2 with basis ℓ and (1/p)ℓ(t^p) [22, Theorem 5.3.3].
    Used in Section 4 to guarantee λp, µp exist in Z_p.
  • standard math Lang's theorems X.4.2 and X.7.3: E_{p-1} and E_{p+1} are weighted polynomials in E4,E6 that are relatively prime modulo p.
    Core input for the non-vanishing of E_{p+1} at supersingular reductions.
  • standard math Mordell's congruence ((p-1)/2)! ≡ (-1)^{(1+h(-p))/2} (mod p) for p ≡ 3 mod 4, p > 3.
    Used in Theorem 2 to evaluate Γ_p(1/2).

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Pith. "Pith review of Cusp forms as $p$-adic limits circumventing $p$-adic version of the Legendre period relation." pith.science (2026). https://pith.science/paper/WO52OUCL

@misc{pith2026250607107,
  author       = {Pith},
  title        = {Pith review of: Cusp forms as $p$-adic limits circumventing $p$-adic version of the Legendre period relation},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WO52OUCL}},
  note         = {Machine review of arXiv:2506.07107}
}
abstract

Several authors have recently proved results which express a cusp form as a $p$-adic limit of weakly holomorphic modular forms under repeated application of Atkin's $U$-operator. Initially, these results had a deficiency: one could not rule out the possibility when a certain quantity vanishes and the final result fails to be true. Later on, Ahlgren and Samart \cite{AS} found a method to prove that no exceptions happen in the specific case considered by El-Guindy and Ono, Hanson and Jameson, and (independently) Dicks. generalized this method to finitely many other cases. In this paper, we present a different approach which allows us to prove a similar non-vanishing result for an infinite family of similar cases. Our approach also allows us to return back to the original example considered by El-Guindy and Ono, where we calculate the (manifestly non-zero) quantity explicitly in terms of Morita's $p$-adic $\Gamma$-function.

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