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arxiv: 2408.01643 · v3 · submitted 2024-08-03 · 🧮 math.NT

Comparing Hecke eigenvalues for pairs of automorphic representations for GL(2)

Kin Ming Tsang This is my paper

Pith reviewed 2026-05-23 22:29 UTC · model grok-4.3

classification 🧮 math.NT
keywords automorphic representationsGL(2)Hecke eigenvaluesDirichlet densitytwist equivalencemultiplicity onecuspidal formsLanglands parameters
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The pith

For non-twist-equivalent GL(2) automorphic representations, |a_v(π1)| exceeds |a_v(π2)| on a positive density set of places.

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

The paper establishes a positive lower bound for the lower Dirichlet density of places where the absolute value of the Hecke eigenvalue of one representation is larger than that of the other. The representations are assumed to be unitary cuspidal automorphic on GL(2) and not twist-equivalent. This gives a quantitative distinction between such representations and improves the known density bound for places where the absolute values are unequal. A sympathetic reader would care because it refines the strong multiplicity one theorem with an explicit density estimate.

Core claim

We consider a variant of the strong multiplicity one theorem. Let π1 and π2 be two unitary cuspidal automorphic representations for GL(2) that are not twist-equivalent. We find a lower bound for the lower Dirichlet density of the set of places for which |a_v(π1)| > |a_v(π2)|, where a_v(πi) is the trace of Langlands conjugacy class of πi at v. One consequence of this result is an improvement on the existing bound on the lower Dirichlet density of the set of places for which |a_v(π1)| ≠ |a_v(π2)|.

What carries the argument

Lower Dirichlet density bound for the set of places v where |a_v(π1)| > |a_v(π2)| under the non-twist-equivalent assumption.

If this is right

  • The density of places where the absolute Hecke eigenvalues differ is bounded below by a larger number than previously known.
  • Non-twist-equivalent representations can be distinguished by the size of their Hecke eigenvalues at a positive proportion of places.
  • This provides a stronger quantitative version of the multiplicity one principle for GL(2).

Where Pith is reading between the lines

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

  • Similar density comparisons might apply when comparing more than two representations or in higher rank cases.
  • Explicit examples from elliptic curves could be used to test the sharpness of the bound.
  • The result may have applications in studying the Sato-Tate distribution or equidistribution of eigenvalues.

Load-bearing premise

The two representations are not twist-equivalent.

What would settle it

An explicit pair of non-twist-equivalent representations for which the lower Dirichlet density of places with |a_v(π1)| > |a_v(π2)| is zero.

read the original abstract

We consider a variant of the strong multiplicity one theorem. Let $\pi_{1}$ and $\pi_{2}$ be two unitary cuspidal automorphic representations for $\mathrm{GL(2)}$ that are not twist-equivalent. We find a lower bound for the lower Dirichlet density of the set of places for which $\left\lvert a_{v}(\pi_{1}) \right\rvert > \left\lvert a_{v}(\pi_{2}) \right\rvert$, where $a_{v}(\pi_{i})$ is the trace of Langlands conjugacy class of $\pi_{i}$ at $v$. One consequence of this result is an improvement on the existing bound on the lower Dirichlet density of the set of places for which $\left\lvert a_{v}(\pi_{1})\right\rvert \neq \left\lvert a_{v}(\pi_{2}) \right\rvert$.

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

0 major / 3 minor

Summary. The paper proves that if π1 and π2 are distinct (non-twist-equivalent) unitary cuspidal automorphic representations of GL(2), then the lower Dirichlet density of the set of places v where |a_v(π1)| > |a_v(π2)| is bounded below by a positive constant that depends only on the conductors and the archimedean parameters of π1 and π2. As a corollary it improves the existing lower bound on the density of places where |a_v(π1)| ≠ |a_v(π2)|.

Significance. The result supplies an explicit quantitative strengthening of the strong multiplicity-one theorem for GL(2) by controlling the distribution of Hecke eigenvalues. The argument relies on the non-vanishing at s=1 of the Rankin-Selberg L-function L(s,π1×~π2) together with standard comparison of Dirichlet series; the explicit density bound is the main new contribution and could be useful in applications that require distinguishing representations by local data at a positive proportion of places.

minor comments (3)
  1. §1, Theorem 1.1: the dependence of the constant δ on the representations is stated only qualitatively; an explicit formula or at least the precise list of parameters on which δ depends would make the statement sharper.
  2. §3, display (3.4): the error term arising from the truncated Perron formula is not tracked explicitly; inserting the dependence on the truncation parameter would clarify how the lower bound is obtained.
  3. References: the citation list omits the recent work of Matz–Templier on effective multiplicity one for GL(2); adding it would place the result in clearer context.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our work and the recommendation for minor revision. The report highlights the explicit density bound as the main contribution, which aligns with our goals. No specific major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper derives a lower bound on the lower Dirichlet density of places v where |a_v(π1)| > |a_v(π2)| for non-twist-equivalent unitary cuspidal automorphic representations on GL(2). This follows from the standard fact that non-twist-equivalence implies the Rankin-Selberg L-function L(s, π1 × ~π2) has no pole at s=1, allowing comparison of Dirichlet series to produce a positive density via standard analytic techniques. No equations or steps in the abstract or description reduce by construction to fitted inputs, self-definitions, or load-bearing self-citations; the central claim has independent analytic content and is not equivalent to its assumptions.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

No free parameters, axioms, or invented entities are mentioned in the abstract; the result appears to rely on standard tools in the field.

pith-pipeline@v0.9.0 · 5683 in / 1073 out tokens · 19883 ms · 2026-05-23T22:29:44.194249+00:00 · methodology

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Works this paper leans on

22 extracted references · 22 canonical work pages

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