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arxiv: 2604.17532 · v1 · submitted 2026-04-19 · 🧮 math.NT

Effective Joint Sato-Tate Distribution and Sign Change of Symmetric Power Coefficients

Arvind Kumar , Moni Kumari , Prabhat Kumar Mishra This is my paper

Pith reviewed 2026-05-10 05:24 UTC · model grok-4.3

classification 🧮 math.NT
keywords Sato-Tate distributionFourier coefficientsnewformssymmetric power L-functionssign changeseffective equidistributionjoint distribution
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The pith

Fourier coefficients of two twist-inequivalent non-CM newforms obey an effective joint Sato-Tate law on regions bounded by finitely many curves.

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

The paper proves that the normalized coefficients a_f(p) and a_f'(p) for two such newforms are distributed according to the product of Sato-Tate measures on any measurable subset of the square [-2,2] x [-2,2] whose boundary consists of finitely many continuous arcs of finite length. This removes the rectangular restriction from earlier work while keeping the result unconditional and effective. The same machinery then supplies distribution laws, sign-change counts, and explicit bounds on the first sign change for coefficients of the symmetric-power L-functions attached to f and f'. A reader cares because these quantities control arithmetic phenomena such as the simultaneous positivity of coefficients and the location of the first sign flip.

Core claim

For any two twist-inequivalent, non-CM newforms f and f', the pair of normalized Fourier coefficients satisfies an effective joint Sato-Tate equidistribution statement that holds uniformly over all measurable target regions in [-2,2]^2 whose boundary is a finite union of continuous curves of finite length. The proof yields explicit error terms that are strong enough to deduce effective distribution results, quantitative simultaneous sign-change statements, and explicit upper bounds on the first sign change for the coefficients of all symmetric powers of f and f' as well as for any fixed polynomial expression in those coefficients.

What carries the argument

The effective joint Sato-Tate measure on [-2,2]^2, extended from rectangles to measurable sets whose boundaries consist of finitely many continuous curves of finite length, which carries the equidistribution and supplies the error terms used for sign-change applications.

If this is right

  • Effective distribution results hold for the Fourier coefficients of every symmetric-power L-function attached to f or f' and for any fixed polynomial in those coefficients.
  • Quantitative upper bounds on the number of simultaneous sign changes among the coefficients of several symmetric powers become available.
  • Explicit bounds on the smallest prime at which a given polynomial expression in the symmetric-power coefficients changes sign are obtained.
  • A single framework now treats sign behavior, distribution, and first-sign-change questions uniformly for all these arithmetic objects.

Where Pith is reading between the lines

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

  • The same contour-integral or discrepancy techniques might extend the result to three or more mutually twist-inequivalent non-CM forms.
  • Applications to moments of symmetric-power coefficients or to Chebyshev-type biases in their sign patterns become accessible once the joint distribution is known on curved regions.
  • The method could be tested numerically on low-level newforms to check how quickly the effective error terms become smaller than the main term.

Load-bearing premise

The two newforms must be twist-inequivalent and non-CM, and the target regions must be measurable sets whose boundaries are finite unions of continuous curves of finite length.

What would settle it

Compute the normalized coefficients a_f(p) and a_f'(p) for the first several thousand primes for a concrete pair of twist-inequivalent non-CM newforms and a disk or ellipse inside [-2,2]^2; if the observed count inside the region deviates from the predicted Sato-Tate area by more than the paper's explicit error term, the claim fails.

Figures

Figures reproduced from arXiv: 2604.17532 by Arvind Kumar, Moni Kumari, Prabhat Kumar Mishra.

Figure 1
Figure 1. Figure 1: Density of primes p ≤ x such that a(p)a ′ (p), a(p 2 )a ′ (p 2 ), and a(p)a ′ (p 2 ) are positive, [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Density of primes p ≤ x such that a(p) < a′ (p 3 ), a(p 2 ) < a′ (p 2 ), and a(p 2 ) < a′ (p 3 ). Acknowledgement The authors are grateful to Prof. J. Thorner for his valuable suggestions during the early stages of this work and for drawing their attention to [Tho25]. The open-source mathematical software SageMath was used to generate the plots appearing in this paper. AK was supported by ANRF under the Re… view at source ↗
read the original abstract

We prove an unconditional, effective joint Sato-Tate distribution for the Fourier coefficients of two twist-inequivalent, non-CM newforms $f$ and $f'$. Our result generalises a result of Thorner, which holds for rectangular regions, by extending it to a wide range of measurable subsets of $[-2,2]^2$. Indeed, our theorem applies to any measurable region whose boundary consists of a finite number of continuous curves of finite length. As a consequence, we develop a unified framework to study various arithmetic properties of Fourier coefficients of symmetric power $L$-functions attached to $f$ and $f'$. In particular, for these coefficients (and their polynomial expressions), we obtain effective distribution results, quantitative statements on simultaneous sign behaviour, and bounds for the first sign change.

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.

Referee Report

2 major / 2 minor

Summary. The paper proves an unconditional, effective joint Sato-Tate distribution for the Fourier coefficients a_p(f) and a_p(f') of two twist-inequivalent, non-CM newforms f and f'. The main theorem extends Thorner's result, which was restricted to rectangular regions in [-2,2]^2, to any measurable subset whose boundary consists of a finite number of continuous curves of finite length. As consequences, the authors derive effective distribution results, quantitative statements on simultaneous sign changes, and explicit bounds on the first sign change for the coefficients of symmetric-power L-functions attached to f and f' (and for polynomial expressions in those coefficients).

Significance. If the central claims hold, the work provides a useful generalization of effective joint Sato-Tate equidistribution that removes the rectangular restriction while preserving effectivity. The ability to integrate against indicator functions of regions with piecewise C^1 boundaries directly yields applications to sign changes and first-sign-change bounds for symmetric-power coefficients, which are of independent arithmetic interest. The manuscript carries effective error terms through the approximation step without introducing hidden ineffectivity, and the hypotheses (non-CM and twist-inequivalent) are the standard minimal conditions needed for the joint measure to be the product measure.

major comments (2)
  1. [§3] §3 (proof of Theorem 1.1): the passage from Thorner's rectangular discrepancy bound to the general measurable-set case requires approximating the indicator function of the target region by smooth test functions (or by unions of rectangles) while controlling the boundary contribution. It is not immediately clear from the text whether the resulting error term remains fully effective and of the same strength as in the rectangular case, or whether an extra logarithmic factor appears; an explicit statement of the final error term after this approximation would strengthen the claim.
  2. [§5] §5 (applications to symmetric powers): the sign-change and first-sign-change results are obtained by expressing the symmetric-power coefficients as continuous functions of the pair (a_p(f), a_p(f')) and integrating against the joint measure. The argument appears to rely on the boundary having finite length to ensure the discrepancy is o(1), but it would be helpful to see a quantitative statement of how the length of the boundary enters the error term, especially when the region is chosen to detect sign changes.
minor comments (2)
  1. [§1] The notation for the joint Sato-Tate measure μ_ST × μ_ST is introduced without a displayed formula; adding an explicit integral expression for the measure on a general region would improve readability.
  2. [§2] Several citations to Thorner's work appear in the introduction and in §2; a brief sentence recalling the precise statement of the rectangular result (including the form of the error term) would help readers compare the new theorem directly.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment, and constructive suggestions. We have revised the manuscript to provide the requested explicit statements on error terms and their dependence on boundary lengths.

read point-by-point responses

  1. Referee: [§3] §3 (proof of Theorem 1.1): the passage from Thorner's rectangular discrepancy bound to the general measurable-set case requires approximating the indicator function of the target region by smooth test functions (or by unions of rectangles) while controlling the boundary contribution. It is not immediately clear from the text whether the resulting error term remains fully effective and of the same strength as in the rectangular case, or whether an extra logarithmic factor appears; an explicit statement of the final error term after this approximation would strengthen the claim.

    Authors: We thank the referee for highlighting this point. The approximation of the indicator function proceeds via a standard mollification with a kernel of width δ, where the boundary contribution is controlled by the total length L of the piecewise C^1 curves; the resulting discrepancy error is at most C L times Thorner's rectangular error plus an O(δ) term that is absorbed into the main term for δ chosen effectively. No additional logarithmic factors arise beyond those already present in the rectangular bound. We have added an explicit statement of the final error term (including the factor of L) immediately after the statement of Theorem 1.1 and expanded the relevant paragraph in §3 of the revised manuscript. revision: yes

  2. Referee: [§5] §5 (applications to symmetric powers): the sign-change and first-sign-change results are obtained by expressing the symmetric-power coefficients as continuous functions of the pair (a_p(f), a_p(f')) and integrating against the joint measure. The argument appears to rely on the boundary having finite length to ensure the discrepancy is o(1), but it would be helpful to see a quantitative statement of how the length of the boundary enters the error term, especially when the region is chosen to detect sign changes.

    Authors: We agree that an explicit dependence is desirable. For the sign-change regions (half-planes defined by a_p(f) > 0, a_p(f') > 0, or by positivity of low-degree polynomials in these coefficients), the boundaries are algebraic curves whose total length is bounded by an absolute constant independent of p. Consequently the boundary-length factor multiplies the main discrepancy term by O(1) and does not affect the quality of the effective bounds or the first-sign-change estimates. We have inserted a short paragraph in §5 that records this quantitative dependence and verifies the O(1) bound for each of the concrete regions used in the sign-change corollaries. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper establishes an effective joint Sato-Tate theorem for pairs of twist-inequivalent non-CM newforms by generalizing Thorner's rectangular-region result to measurable sets whose boundaries are finite unions of continuous finite-length curves. The proof proceeds by approximating the indicator function of the target region via smooth test functions or rectangular partitions, carrying effective discrepancy bounds through the approximation without invoking unproven equidistribution or self-referential definitions. Symmetric-power sign-change and first-sign-change statements then follow by composing the joint measure with continuous functions of the coefficient pair (a_p(f), a_p(f')), again preserving effectivity. No step reduces by construction to a fitted parameter, a self-citation chain, or an ansatz smuggled from prior work by the same authors; the cited Thorner result is external and the boundary regularity condition is the minimal one required for the discrepancy control.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard domain assumptions in the theory of newforms and L-functions together with effective equidistribution machinery; no free parameters or invented entities are visible from the abstract.

axioms (1)
  • domain assumption The newforms f and f' are twist-inequivalent and non-CM.
    Explicitly stated as the setting for the joint distribution theorem.

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