Year: 2025
Type: preprint
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-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 $1728$.
Source: arXiv (Cornell University)
Author: Gomez, Kevin
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7084447508
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