Dimension formulas for modular form spaces of rational weights, the classification of eta-quotient characters and an extension of Martin's theorem
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866911974458130432 |
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| author | Zhu, Xiao-Jie |
| author_facet | Zhu, Xiao-Jie |
| contents | We give an explicit formula for dimensions of spaces of rational-weight modular forms whose multiplier systems are induced by eta-quotients of fractional exponents. As the first application, we give series expressions of Fourier coefficients of the $n$-th root of certain infinite $q$-products. As the second application, we extend Yves Martin's list of multiplicative holomorphic eta-quotients of integral weights by first extending the meaning of multiplicativity, then identifying one-dimensional spaces, and finally applying Wohlfahrt's extension of Hecke operators. A table containing $2277$ of such eta-quotients is presented. As a related result, we completely classify the multiplier systems induced by eta-quotients of integral exponents. For instance, there are totally $384$ such multiplier systems on $Γ_0(4)$ for any fixed weight. We also provide SageMath programs on checking the theorems and generating the tables. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_00246 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Dimension formulas for modular form spaces of rational weights, the classification of eta-quotient characters and an extension of Martin's theorem Zhu, Xiao-Jie Number Theory 11F12 (Primary) 11F20, 11F25, 11F30, 11L05, 30F10 (Secondary) We give an explicit formula for dimensions of spaces of rational-weight modular forms whose multiplier systems are induced by eta-quotients of fractional exponents. As the first application, we give series expressions of Fourier coefficients of the $n$-th root of certain infinite $q$-products. As the second application, we extend Yves Martin's list of multiplicative holomorphic eta-quotients of integral weights by first extending the meaning of multiplicativity, then identifying one-dimensional spaces, and finally applying Wohlfahrt's extension of Hecke operators. A table containing $2277$ of such eta-quotients is presented. As a related result, we completely classify the multiplier systems induced by eta-quotients of integral exponents. For instance, there are totally $384$ such multiplier systems on $Γ_0(4)$ for any fixed weight. We also provide SageMath programs on checking the theorems and generating the tables. |
| title | Dimension formulas for modular form spaces of rational weights, the classification of eta-quotient characters and an extension of Martin's theorem |
| topic | Number Theory 11F12 (Primary) 11F20, 11F25, 11F30, 11L05, 30F10 (Secondary) |
| url | https://arxiv.org/abs/2408.00246 |