Hecke polynomials for the mock modular form arising from the Delta-function
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914014911528960 |
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| author | Gomez, Kevin Ono, Ken |
| author_facet | Gomez, Kevin Ono, Ken |
| contents | We consider a mock modular form $M_Δ(τ)$ that arises naturally from Ramanujan's Delta-function. It is a weight $-10$ harmonic Maass form whose nonholomorphic part is the "period integral function'' of $Δ(τ)$. The Hecke operator $T_{-10}(m)$ acts on this mock modular form in terms of Ramanujan's $τ(m)$ and a monic degree $m$ polynomial $F_m(x),$ evaluated at $x=j(τ).$ In analogy with results by Asai, Kaneko, and Ninomiya on the zeros of Hecke polynomials for the $j$-function, we prove that the zeros of each $F_m(x)$, including $x=0$ and $x=1728,$ are distinct and lie in $[0, 1728]$. Additionally, as $m \to +\infty,$ these zeros become equidistributed in $[0, 1728].$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_17178 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hecke polynomials for the mock modular form arising from the Delta-function Gomez, Kevin Ono, Ken Number Theory 11F30, 11F25 We consider a mock modular form $M_Δ(τ)$ that arises naturally from Ramanujan's Delta-function. It is a weight $-10$ harmonic Maass form whose nonholomorphic part is the "period integral function'' of $Δ(τ)$. The Hecke operator $T_{-10}(m)$ acts on this mock modular form in terms of Ramanujan's $τ(m)$ and a monic degree $m$ polynomial $F_m(x),$ evaluated at $x=j(τ).$ In analogy with results by Asai, Kaneko, and Ninomiya on the zeros of Hecke polynomials for the $j$-function, we prove that the zeros of each $F_m(x)$, including $x=0$ and $x=1728,$ are distinct and lie in $[0, 1728]$. Additionally, as $m \to +\infty,$ these zeros become equidistributed in $[0, 1728].$ |
| title | Hecke polynomials for the mock modular form arising from the Delta-function |
| topic | Number Theory 11F30, 11F25 |
| url | https://arxiv.org/abs/2506.17178 |