Hecke Eigenvalues of Ikeda Lifts

Ameya Pitale; Nagarjuna Chary Addanki

arxiv: 2605.16083 · v2 · pith:W5RAJ4XLnew · submitted 2026-05-15 · 🧮 math.NT

Hecke Eigenvalues of Ikeda Lifts

Nagarjuna Chary Addanki , Ameya Pitale This is my paper

Pith reviewed 2026-05-19 18:47 UTC · model grok-4.3

classification 🧮 math.NT

keywords Hecke eigenvaluesIkeda liftssymplectic groupautomorphic formspositivityHecke algebra

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The pith

Hecke eigenvalues of Ikeda lifts are positive for all sufficiently large primes.

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

The paper derives an explicit formula for the Hecke eigenvalues λ_F(p^r) of Ikeda lifts by applying the spherical map to the Hecke algebra of the symplectic group. This formula allows the eigenvalues to be expressed as a polynomial in powers of p to the plus or minus one half, with a positive leading term and bounded coefficients. As a result, these eigenvalues are positive when p is large enough. A sympathetic reader would care because this provides concrete information on the signs of eigenvalues associated with these automorphic forms.

Core claim

Using the spherical map for the Hecke algebra of the symplectic group, the authors obtain an explicit formula for the eigenvalues λ_F(p^r) of Ikeda lifts. From this formula, they show that λ_F(p^r) can be written as a polynomial in p^{±1/2} with a positive leading term. The coefficients of this polynomial are bounded, and therefore the Hecke eigenvalues λ_F(p^r) are positive for all sufficiently large primes p.

What carries the argument

The spherical map for the Hecke algebra of the symplectic group, which transfers the Hecke eigenvalues from the Ikeda lift to an explicit algebraic expression.

If this is right

λ_F(p^r) can be expressed as a polynomial in p^{±1/2} with positive leading term.
The coefficients of this polynomial are bounded.
λ_F(p^r) is positive for all sufficiently large primes p.

Where Pith is reading between the lines

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

This positivity might help establish non-vanishing results for associated L-functions at special points.
The same approach could extend to other lifts or to higher powers in related groups.

Load-bearing premise

The spherical map for the Hecke algebra of the symplectic group correctly transfers the Hecke eigenvalues from the Ikeda lift to an explicit algebraic expression that can be analyzed as a polynomial.

What would settle it

Computing the Hecke eigenvalues λ_F(p^r) explicitly for a known Ikeda lift and a large prime p, and checking whether the value is positive or not.

read the original abstract

In this paper, we study the Hecke eigenvalues of Ikeda lifts. Using the spherical map for the Hecke algebra of the symplectic group, we obtain an explicit formula for the eigenvalues $\lambda_F(p^r)$. From this formula, we show that $\lambda_F(p^r)$ can be written as a polynomial in $p^{\pm 1/2}$ with a positive leading term. Furthermore, we prove that the coefficients of this polynomial are bounded and, as a consequence, the Hecke eigenvalues $\lambda_F(p^r)$ are positive for all sufficiently large primes $p$.

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

The paper derives an explicit polynomial formula for Hecke eigenvalues of Ikeda lifts via the spherical map and proves eventual positivity from bounded coefficients and a positive leading term.

read the letter

The main point is that the authors produce an explicit formula for λ_F(p^r) on Ikeda lifts by applying the spherical map to the Hecke algebra, write the result as a polynomial in p to plus or minus one-half with positive leading term, and then bound the remaining coefficients to get positivity for all large primes p. This is a concrete algebraic description rather than an existence statement, which is the useful part of the work. The derivation follows the usual path from the map to the polynomial expression, and the positivity step is a direct consequence once the bounds are in hand. That kind of explicit control is helpful inside the subfield because it lets people compute signs or check small cases without extra machinery. The argument looks internally consistent and does not rely on fitted parameters or circular definitions. One minor soft spot is that the coefficient bounds are stated without visible dependence on the weight or level of the lift; if those bounds grow, the threshold for “sufficiently large” p could become impractical for some families, though the abstract does not indicate any such growth. The paper also does not include numerical checks against known low-weight examples, which would have made the boundedness claim easier to verify quickly. Overall this is a standard, focused computation in the theory of Siegel modular forms and their lifts. It is aimed at readers already working with automorphic forms on GSp(2n) or with sign questions for Hecke eigenvalues. A specialist in that area would get direct value from the formula for further calculations. The work is grounded enough and the claim is specific enough that it deserves a serious referee rather than a desk rejection.

Referee Report

0 major / 2 minor

Summary. The manuscript derives an explicit formula for the Hecke eigenvalues λ_F(p^r) of Ikeda lifts by applying the spherical map to the Hecke algebra of the symplectic group. It then rewrites this formula as a polynomial in p^{±1/2} possessing a positive leading term whose coefficients are bounded, from which positivity of λ_F(p^r) for all sufficiently large primes p is deduced.

Significance. If the derivation is correct, the result supplies a concrete, analyzable expression for the Hecke eigenvalues of Ikeda lifts and establishes their eventual positivity. This contributes to the study of eigenvalue distributions for Siegel modular forms and may aid in applications involving L-functions or bounds on automorphic representations. The use of the spherical map to reach a parameter-free polynomial form is a methodological strength.

minor comments (2)

The transition from the spherical map to the explicit polynomial expression would be clearer if a short verification for small values of r (e.g., r=1 or r=2) were included to confirm the leading coefficient and boundedness.
The introduction would benefit from a brief comparison with existing results on Hecke eigenvalues of other lifts (such as Saito-Kurokawa or Maass lifts) to better situate the novelty of the positivity statement.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and for recommending minor revision. The referee's summary accurately captures our use of the spherical map to derive an explicit formula for the Hecke eigenvalues of Ikeda lifts, the rewriting as a polynomial in p^{±1/2} with positive leading term and bounded coefficients, and the resulting positivity statement for large primes. We are glad that the potential contributions to eigenvalue distributions and L-function applications are noted.

Circularity Check

0 steps flagged

No significant circularity identified

full rationale

The derivation begins with the spherical map for the Hecke algebra of the symplectic group to produce an explicit formula for λ_F(p^r). This formula is then rewritten as a polynomial in p^{±1/2} whose leading term is positive and whose remaining coefficients are shown to be bounded, implying positivity for all sufficiently large p. None of these steps reduces by construction to a fitted parameter, a self-referential definition, or a load-bearing self-citation. The explicit algebraic expression and the subsequent polynomial analysis constitute independent mathematical content that does not presuppose the final positivity statement.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard facts about Hecke algebras and the definition of Ikeda lifts; no free parameters, ad-hoc axioms, or new invented entities are mentioned in the abstract.

axioms (1)

domain assumption The spherical map for the Hecke algebra of the symplectic group produces the correct eigenvalues for the Ikeda lift.
Invoked to obtain the explicit formula for λ_F(p^r).

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discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Using the spherical map for the Hecke algebra of the symplectic group, we obtain an explicit formula for the eigenvalues λ_F(p^r). From this formula, we show that λ_F(p^r) can be written as a polynomial in p^{±1/2} with a positive leading term.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

λ_F(p^r) = p^{r(nk - n/2)} ∑ ... φ_{2n}(p^{-1}) / (φ_{k1}(p^{-1}) ... φ_{kt}(p^{-1}))

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extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.