Zeros of Hecke polynomials arising from weak eigenforms

TL;DR

This work extends the zero-distribution phenomenon of divisor polynomials from holomorphic modular forms to a broad class of harmonic Maass forms. It defines Hecke polynomials $P_n(F;x)$ for weak Hecke eigenforms $F$ of weight $2-k$ and proves that, for large $n$, their zeros are simple, lie in $[0,1728]$, and become equidistributed as $n\to\infty$. The strategy represents $F$ as a linear combination of Maass–Poincaré series and analyzes Hecke action along the arc $\mathcal{A}$, isolating a dominant cosine term and bounding tails with Maass–Poincaré theory and Bessel bounds. The results yield a clean degree formula $mn-b(k-2)$ for $P_n(F;x)$ and characterize possible zeros at the cusps $0$ and $1728$ depending on $k \bmod 12$, offering a robust generalization of the Rankin–Swinnerton-Dyer and Asai–Kaneko–Ninomiya picture to harmonic Maass contexts and establishing equidistribution in a natural interval.

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, lies in $[0,1728]$, and the zeros equidistribute on this interval. 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-Poincaré 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 $1728$.

Theorems & Definitions (19)

• Theorem 1.1
• Theorem 1.2
• Remark
• Corollary 1.3
• proof : Proof of Theorem \ref{['Theorem1']}
• Lemma 4.1: Theorem 6.10 of HMF
• Lemma 4.2
• proof
• Lemma 4.3
• proof