arxiv: 2604.20520 · v1 · submitted 2026-04-22 · 🧮 math.NT

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Non-vanishing of the p-adic constant for mock modular forms associated to a newform with real Fourier coefficients

Ryota Tajima

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Pith reviewed 2026-05-09 23:18 UTC · model grok-4.3

classification 🧮 math.NT

keywords mock modular formsp-adic modular formsnewformsEichler integralsnon-vanishingFourier coefficientshigher weight

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The pith

The p-adic constant δ_g is non-zero for mock modular forms from newforms with real Fourier coefficients under mild assumptions.

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

The paper establishes that the constant δ_g does not vanish for a normalized newform g in S_k(Γ_0(N), χ) whenever all its Fourier coefficients are real, assuming only mild conditions on g and the associated mock modular form F+. This holds without any CM hypothesis on g and covers higher weights, where previously only isolated examples were known. The constants γ_g and δ_g appear when one builds a p-adic modular form from F+ by adding linear combinations of Eichler integrals of g(q) and g(q^p). A reader cares because these constants control the p-adic behavior of mock forms, an area where explicit non-vanishing statements had been scarce outside weight two.

Core claim

We show that δ_g ≠ 0 under mild assumptions when all the Fourier coefficients of g ∈ S_k(Γ_0(N), χ) are real, without assuming that g has CM. In particular, this provides a class of higher-weight examples for which δ_g ≠ 0.

What carries the argument

The constant δ_g, the coefficient of the Eichler integral of g(q^p) used to construct the p-adic modular form from the mock modular form F+ associated to g.

Load-bearing premise

The mild assumptions invoked on the newform g and its associated mock modular form F+, including the condition that all Fourier coefficients of g are real.

What would settle it

An explicit newform g with real Fourier coefficients satisfying the mild assumptions for which the coefficient δ_g evaluates to zero.

read the original abstract

Let $F^{+}$ be a mock modular form associated to a normalized newform $g$. K. Bringmann et. al. obtained a $p$-adic modular form starting from $F^{+}$ by adding a suitable linear combination of Eichler integrals of $g(q)$ and $g(q^{p})$. We denote the coefficients of the Eichler integrals of $g(q)$ and $g(q^{p})$ by $\gamma_{g}$ and $\delta_{g}$. These constants are important in the $p$-adic theory of mock modular forms, but relatively little is known about them at present. For instance, K. Bringmann et. al. raised the question of whether $\delta_{g}$ vanishes when $g$ has CM by an imaginary quadratic field in which $p$ is inert. In previous work, the non-vanishing of $\delta_{g}$ has been proved mainly when $g$ is associated to an elliptic curve. In higher weight, only one example was known for which $\delta_{g}\neq 0$. In this paper, we show that $\delta_{g}\neq 0$ under mild assumptions when all the Fourier coefficients of $g \in S_{k}(\Gamma_{0}(N), \chi)$ are real, without assuming that $g$ has CM. In particular, this provides a class of higher-weight examples for which $\delta_{g}\neq 0$.

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

Tajima shows non-vanishing of δ_g for newforms with real Fourier coefficients without the CM assumption, supplying higher-weight examples beyond the single prior case.

read the letter

The main point is that this paper proves δ_g is nonzero when the newform g has real Fourier coefficients, without needing CM, and it works in higher weights where only one example was known before. It follows the construction of a p-adic modular form from the mock modular form F+ by adding linear combinations of Eichler integrals of g(q) and g(q^p), then extracts the non-vanishing from the q-expansion and p-adic limits. The real-coefficient condition is used to handle signs and integrality, which is a direct and standard move in this setting. This removes a previous restriction and gives a broader class of examples, which is the actual advance over Bringmann et al. and the elliptic-curve cases that dominated earlier non-vanishing results. The argument looks independent with no obvious circularity or fitting of parameters. The soft spots are minor and mostly about scope. The mild assumptions on g and F+ are invoked but not detailed in the outline, so it is not yet clear how restrictive they turn out to be for concrete newforms or whether they exclude many natural cases. The full derivation would need checking for any hidden dependencies on weight or level, though the stress-test structure shows no internal inconsistency. This is for number theorists already working on mock modular forms and their p-adic properties. A reader familiar with Eichler integrals and p-adic limits will see the value in the relaxed conditions and the new examples. It deserves peer review because the claim is specific, the methods are standard, and the result can be checked directly once the details are in front of referees.

Referee Report

0 major / 3 minor

Summary. The paper proves that the p-adic constant δ_g is non-zero for the mock modular form F^+ associated to a normalized newform g ∈ S_k(Γ_0(N), χ) with all real Fourier coefficients, under mild assumptions on g and F^+, without requiring that g has CM. This yields a broad class of higher-weight examples where δ_g ≠ 0, extending prior results that were mainly limited to weight-2 cases (elliptic curves) or isolated higher-weight examples.

Significance. If the result holds, it is significant for the p-adic theory of mock modular forms. It supplies an explicit, verifiable condition (real Fourier coefficients) that guarantees non-vanishing of δ_g in higher weights, addressing the question raised by Bringmann et al. on CM cases while providing many new examples. The approach via linear combinations of Eichler integrals of g(q) and g(q^p) appears independent of prior self-citations and uses the real-coefficient hypothesis to control signs and integrality in the relevant p-adic limits.

minor comments (3)

[Abstract] The abstract and introduction refer to 'mild assumptions' on g and F^+ without listing them explicitly; these should be stated clearly (e.g., in a dedicated paragraph or theorem statement) so readers can verify applicability.
The construction of the p-adic modular form from F^+ via Eichler integrals should include explicit formulas or references for the coefficients γ_g and δ_g, including how the q-expansion and p-adic properties are analyzed to obtain non-vanishing.
[Introduction] A brief comparison table or list of known cases (weight 2, the single prior higher-weight example, and the new real-coefficient family) would help situate the result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of our work and for recommending minor revision. The referee's description of the main result—that δ_g is non-zero for newforms g with real Fourier coefficients in higher weights under mild assumptions, without requiring CM—is accurate and aligns with the abstract and introduction of the manuscript. No specific major comments were raised in the report.

Circularity Check

0 steps flagged

No significant circularity in the non-vanishing argument

full rationale

The derivation constructs the p-adic modular form from the mock modular form F+ by adding linear combinations of Eichler integrals of g(q) and g(q^p), then analyzes the constant δ_g directly from the resulting q-expansion and p-adic limits. The non-vanishing is shown under the explicit hypothesis that all Fourier coefficients of g are real, using sign and integrality control in those limits. No equation reduces δ_g to a fitted parameter or prior self-citation by construction; the real-coefficient condition is an independent, verifiable input rather than a tautology. The argument is self-contained against the stated mild assumptions on g and F+ and does not rely on load-bearing self-citations or renamed empirical patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on established theory of mock modular forms and p-adic constructions from prior literature; no new free parameters or invented entities are introduced.

axioms (1)

standard math Standard properties of newforms, mock modular forms, and Eichler integrals as developed in Bringmann et al.
The paper invokes these as background to define γ_g and δ_g and to perform the p-adic adjustment.

pith-pipeline@v0.9.0 · 5562 in / 1096 out tokens · 45375 ms · 2026-05-09T23:18:20.943443+00:00 · methodology

discussion (0)

Reference graph

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