The Explicit Hypergeometric-Modularity Method II
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2024
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| _version_ | 1866929601109819392 |
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| author | Allen, Michael Grove, Brian Long, Ling Tu, Fang-Ting |
| author_facet | Allen, Michael Grove, Brian Long, Ling Tu, Fang-Ting |
| contents | In the first paper of this sequence, we provided an explicit hypergeometric modularity method by combining different techniques from the classical, $p$-adic, and finite field settings. In this article, we explore an application of this method from a motivic viewpoint through some known hypergeometric well-poised formulae of Whipple and McCarthy. We first use the method to derive a class of special weight three modular forms, labeled as $\mathbb{K}_2$-functions. Then using well-poised hypergeometric formulae we further construct a class of degree four Galois representations of the absolute Galois groups of the corresponding cyclotomic fields. These representations are then shown to be extendable to $G_{\mathbb{Q}}$ and the $L$-function of each extension coincides with the $L$-function of an automorphic form. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2411_15116 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Explicit Hypergeometric-Modularity Method II Allen, Michael Grove, Brian Long, Ling Tu, Fang-Ting Number Theory In the first paper of this sequence, we provided an explicit hypergeometric modularity method by combining different techniques from the classical, $p$-adic, and finite field settings. In this article, we explore an application of this method from a motivic viewpoint through some known hypergeometric well-poised formulae of Whipple and McCarthy. We first use the method to derive a class of special weight three modular forms, labeled as $\mathbb{K}_2$-functions. Then using well-poised hypergeometric formulae we further construct a class of degree four Galois representations of the absolute Galois groups of the corresponding cyclotomic fields. These representations are then shown to be extendable to $G_{\mathbb{Q}}$ and the $L$-function of each extension coincides with the $L$-function of an automorphic form. |
| title | The Explicit Hypergeometric-Modularity Method II |
| topic | Number Theory |
| url | https://arxiv.org/abs/2411.15116 |