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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2408.11473 |
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| _version_ | 1866909292959891456 |
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| author | Armana, Cécile |
| author_facet | Armana, Cécile |
| contents | We study two analogs, for modular forms over $\mathbb{F}_{q}(T)$, of the pairing between Hecke algebra and cusp forms given by the first coefficient in the expansion. For Drinfeld modular forms, the $\mathbb{C}_{\infty}$-pairing is provided by the first coefficient of their $t$-expansion at infinity. For $\mathbb{Z}$-valued harmonic cochains, the $\mathbb{Z}$-pairing is given by their Fourier coefficient with respect to the trivial ideal. We prove that, contrarily to classical cusp forms, both pairings in weight $2$ are not perfect in a quite general setting, namely for the congruence subgroup $Γ_0(\mathfrak{n})$ with any prime ideal $\mathfrak{n}$ in $\mathbb{F}_{q}[T]$ of degree $\geq 5$. We show it by exhibiting a common element of the Hecke algebra in the kernels of both pairings and proving that it is non-zero using computations with modular symbols over $\mathbb{F}_{q}(T)$. Finally we present computational data on other kernel elements of these pairings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_11473 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Non-perfect pairings between Hecke algebra and modular forms over function fields Armana, Cécile Number Theory We study two analogs, for modular forms over $\mathbb{F}_{q}(T)$, of the pairing between Hecke algebra and cusp forms given by the first coefficient in the expansion. For Drinfeld modular forms, the $\mathbb{C}_{\infty}$-pairing is provided by the first coefficient of their $t$-expansion at infinity. For $\mathbb{Z}$-valued harmonic cochains, the $\mathbb{Z}$-pairing is given by their Fourier coefficient with respect to the trivial ideal. We prove that, contrarily to classical cusp forms, both pairings in weight $2$ are not perfect in a quite general setting, namely for the congruence subgroup $Γ_0(\mathfrak{n})$ with any prime ideal $\mathfrak{n}$ in $\mathbb{F}_{q}[T]$ of degree $\geq 5$. We show it by exhibiting a common element of the Hecke algebra in the kernels of both pairings and proving that it is non-zero using computations with modular symbols over $\mathbb{F}_{q}(T)$. Finally we present computational data on other kernel elements of these pairings. |
| title | Non-perfect pairings between Hecke algebra and modular forms over function fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2408.11473 |