Eichler-Selberg relations for singular moduli
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866911927552180224 |
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| author | Deng, Yuqi Matsusaka, Toshiki Ono, Ken |
| author_facet | Deng, Yuqi Matsusaka, Toshiki Ono, Ken |
| contents | The Eichler-Selberg trace formula expresses the trace of Hecke operators on spaces of cusp forms as weighted sums of Hurwitz-Kronecker class numbers. We extend this formula to a natural class of relations for traces of singular moduli, where one views class numbers as traces of the constant function $j_0(τ)=1$. More generally, we consider the singular moduli for the Hecke system of modular functions \[ j_m(τ) := mT_m \left(j(τ)-744\right). \] For each $ν\geq 0$ and $m\geq 1$, we obtain an Eichler-Selberg relation. For $ν=0$ and $m\in \{1, 2\},$ these relations are Kaneko's celebrated singular moduli formulas for the coefficients of $j(τ).$ For each $ν\geq 1$ and $m\geq 1,$ we obtain a new Eichler-Selberg trace formula for the Hecke action on the space of weight $2ν+2$ cusp forms, where the traces of $j_m(τ)$ singular moduli replace Hurwitz-Kronecker class numbers. These formulas involve a new term that is assembled from values of symmetrized shifted convolution $L$-functions. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14280 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Eichler-Selberg relations for singular moduli Deng, Yuqi Matsusaka, Toshiki Ono, Ken Number Theory 11F37, 11F50, 11F67 The Eichler-Selberg trace formula expresses the trace of Hecke operators on spaces of cusp forms as weighted sums of Hurwitz-Kronecker class numbers. We extend this formula to a natural class of relations for traces of singular moduli, where one views class numbers as traces of the constant function $j_0(τ)=1$. More generally, we consider the singular moduli for the Hecke system of modular functions \[ j_m(τ) := mT_m \left(j(τ)-744\right). \] For each $ν\geq 0$ and $m\geq 1$, we obtain an Eichler-Selberg relation. For $ν=0$ and $m\in \{1, 2\},$ these relations are Kaneko's celebrated singular moduli formulas for the coefficients of $j(τ).$ For each $ν\geq 1$ and $m\geq 1,$ we obtain a new Eichler-Selberg trace formula for the Hecke action on the space of weight $2ν+2$ cusp forms, where the traces of $j_m(τ)$ singular moduli replace Hurwitz-Kronecker class numbers. These formulas involve a new term that is assembled from values of symmetrized shifted convolution $L$-functions. |
| title | Eichler-Selberg relations for singular moduli |
| topic | Number Theory 11F37, 11F50, 11F67 |
| url | https://arxiv.org/abs/2406.14280 |