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| Format: | Preprint |
| Publié: |
2024
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| Accès en ligne: | https://arxiv.org/abs/2412.07054 |
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| _version_ | 1866911158066216960 |
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| author | Rosen, Esme |
| author_facet | Rosen, Esme |
| contents | Recently, Allen, Grove, Long, and Tu proposed an explicit Hypergeometric-Modularity method which gives a concrete link between certain hypergeometric objects and modular forms. The theory is exemplified by a collection of 199 weight 3 modular forms. Among other properties their process shows that the $L$-value of such a modular form at 1 is an explicit multiple of a ${}_3F_2(1)$ hypergeometric series. Using the framework of a finite Coxeter group governing the invariance group of normalized ${}_3F_2(1)$ series, this paper fully classifies and describes the possible Hecke eigenforms whose $L$-values that can be obtained using this method. In addition, we determine when these modular forms differ by twist of a finite-order character using the perspective of hypergeometric functions. As one application, we reinterpret a classical identity of hypergeometric series as a formula involving $L$-values of two Hecke eigenforms that differ by a twist. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_07054 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | $L$-values of certain weight 3 Modular Forms and Transformations of Hypergeometric Series Rosen, Esme Number Theory 11J91 (primary), 11F67 (secondary) Recently, Allen, Grove, Long, and Tu proposed an explicit Hypergeometric-Modularity method which gives a concrete link between certain hypergeometric objects and modular forms. The theory is exemplified by a collection of 199 weight 3 modular forms. Among other properties their process shows that the $L$-value of such a modular form at 1 is an explicit multiple of a ${}_3F_2(1)$ hypergeometric series. Using the framework of a finite Coxeter group governing the invariance group of normalized ${}_3F_2(1)$ series, this paper fully classifies and describes the possible Hecke eigenforms whose $L$-values that can be obtained using this method. In addition, we determine when these modular forms differ by twist of a finite-order character using the perspective of hypergeometric functions. As one application, we reinterpret a classical identity of hypergeometric series as a formula involving $L$-values of two Hecke eigenforms that differ by a twist. |
| title | $L$-values of certain weight 3 Modular Forms and Transformations of Hypergeometric Series |
| topic | Number Theory 11J91 (primary), 11F67 (secondary) |
| url | https://arxiv.org/abs/2412.07054 |