Zeros of Hecke polynomials arising from weak eigenforms
Pith reviewed 2026-05-18 11:28 UTC · model grok-4.3
The pith
Weak Hecke eigenforms of weight 2-k give rise to Hecke polynomials whose zeros become simple and confined to [0,1728] for large n.
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
Core claim
We attach Hecke polynomials P_n(F;x) to weak Hecke eigenforms F of weight 2-k and show that, for large n, every zero is simple and lies in [0,1728]. The construction pulls back a weakly holomorphic Hecke combination of F along j; the analysis follows Hecke orbits on the unit-circle arc A, isolating a dominant cosine term and controlling the tail via Maass-Poincaré series and Whittaker/Bessel bounds.
What carries the argument
Hecke polynomials P_n(F;x) obtained by pulling back along the j-invariant, whose zeros are located by tracking dominant terms in sums over Hecke orbits on the unit circle arc.
If this is right
- A clean formula for the degree of P_n(F;x) follows directly from the construction.
- Simple criteria determine when 0 or 1728 is a zero of the polynomial.
- The result applies to a broad class of harmonic Maass forms extending beyond holomorphic cases.
Where Pith is reading between the lines
- Similar zero confinement might be provable for polynomials arising from other types of automorphic forms using analogous orbit analysis.
- Numerical checks for moderate n could reveal how quickly the zeros enter the interval [0,1728].
Load-bearing premise
The proof relies on isolating a dominant cosine term from the sum over Hecke orbits while bounding the remaining contributions using estimates from Maass-Poincaré series and Whittaker and Bessel functions.
What would settle it
Explicit computation of P_n(F;x) for a concrete weak eigenform F and sufficiently large n, followed by numerical verification that all roots are simple and inside [0,1728].
read the original abstract
We attach Hecke polynomials $P_n(F;x)$ to weak Hecke eigenforms $F$ of weight $2-k$ and show that, for large $n$, every zero is simple and lies in $[0,1728]$. The construction pulls back a weakly holomorphic Hecke combination of $F$ along $j$; the analysis follows Hecke orbits on the unit-circle arc $\mathcal{A}$, isolating a dominant "cosine" term and controlling the tail via Maass-Poincar\'e series and Whittaker/Bessel bounds. This extends the Rankin--Swinnerton-Dyer/Asai--Kaneko--Ninomiya picture from holomorphic forms to a broad class of harmonic Maass forms and yields a clean degree-monicity formula and simple criteria for zeros at $0$ and $1728$.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper attaches Hecke polynomials P_n(F;x) to weak Hecke eigenforms F of weight 2-k and shows that for large n every zero is simple and lies in [0,1728]. The construction pulls back a weakly holomorphic Hecke combination of F along j; the analysis follows Hecke orbits on the unit-circle arc A, isolating a dominant cosine term and controlling the tail via Maass-Poincaré series and Whittaker/Bessel bounds. This extends the Rankin-Swinnerton-Dyer/Asai-Kaneko-Ninomiya picture from holomorphic forms to harmonic Maass forms and yields a degree-monicity formula together with criteria for zeros at 0 and 1728.
Significance. If the analytic estimates hold, the result would extend classical zero-location theorems for Hecke polynomials to a substantially larger class of forms, including weak eigenforms and harmonic Maass forms. The claimed degree-monicity formula and explicit criteria for zeros at the endpoints 0 and 1728 would be useful technical tools. The approach via Maass-Poincaré series for tail control appears consistent with existing methods in the literature on non-holomorphic forms.
major comments (1)
- Abstract: the central claims on simplicity and location of zeros rest on isolating a dominant cosine term along Hecke orbits on arc A and on explicit tail bounds from Maass-Poincaré series together with Whittaker/Bessel estimates; without the full text these estimates, error terms, and the precise definition of the polynomials P_n(F;x) cannot be verified, so the load-bearing analytic steps remain unconfirmed.
Simulated Author's Rebuttal
We thank the referee for their summary and for recognizing the potential significance of extending classical zero-location results to weak eigenforms and harmonic Maass forms. We address the major comment below.
read point-by-point responses
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Referee: Abstract: the central claims on simplicity and location of zeros rest on isolating a dominant cosine term along Hecke orbits on arc A and on explicit tail bounds from Maass-Poincaré series together with Whittaker/Bessel estimates; without the full text these estimates, error terms, and the precise definition of the polynomials P_n(F;x) cannot be verified, so the load-bearing analytic steps remain unconfirmed.
Authors: The abstract summarizes the main results and approach. The full manuscript contains the precise definition of the Hecke polynomials P_n(F;x), the detailed analysis of Hecke orbits on the arc A, the isolation of the dominant cosine term, and the explicit tail bounds derived from Maass-Poincaré series together with Whittaker/Bessel estimates, including all error terms. These appear in the sections following the introduction, where the construction via pullback along j and the subsequent estimates are carried out in full. revision: no
Circularity Check
No significant circularity identified from available text
full rationale
The abstract describes constructing Hecke polynomials P_n(F;x) attached to weak eigenforms and proving simplicity and location of zeros for large n via analysis of Hecke orbits on arc A, dominant cosine isolation, and tail bounds from Maass-Poincaré series plus Whittaker/Bessel estimates. This extends prior work on holomorphic forms but shows no self-definitional reduction, fitted-input-as-prediction, or load-bearing self-citation chain. The derivation is presented as analytic and independent of its own outputs. With only the abstract available, no equations or internal steps can be inspected for equivalence by construction; the claim remains self-contained against external benchmarks.
Axiom & Free-Parameter Ledger
axioms (1)
- standard math Standard bounds on Whittaker and Bessel functions suffice to control the tail of the Fourier expansion along Hecke orbits.
Lean theorems connected to this paper
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IndisputableMonolith/Cost/FunctionalEquation.leanwashburn_uniqueness_aczel unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
isolating a dominant “cosine” term and controlling the tail via Maass–Poincaré series and Whittaker/Bessel bounds
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IndisputableMonolith/Foundation/AlexanderDuality.leanalexander_duality_circle_linking unclear?
unclearRelation between the paper passage and the cited Recognition theorem.
zeros ... lie in [0,1728] ... equidistribute on this interval
What do these tags mean?
- matches
- The paper's claim is directly supported by a theorem in the formal canon.
- supports
- The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
- 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.
discussion (0)
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