Zeros of Hecke polynomials arising from weak eigenforms
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866913028181590016 |
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| author | Gomez, Kevin |
| author_facet | Gomez, Kevin |
| contents | 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 $1728$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_26519 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Zeros of Hecke polynomials arising from weak eigenforms Gomez, Kevin Number Theory 11F30, 11F25 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 $1728$. |
| title | Zeros of Hecke polynomials arising from weak eigenforms |
| topic | Number Theory 11F30, 11F25 |
| url | https://arxiv.org/abs/2509.26519 |