Frobenius structure on hypergeometric equations, p-adic polygamma values and p-adic L-values
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2023
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| _version_ | 1866917510344867840 |
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| author | Asakura, Masanori Hagihara, Kei |
| author_facet | Asakura, Masanori Hagihara, Kei |
| contents | Recently, Kedlaya proves certain formula describing explicitly the Frobenius structure on a hypergeometric equation. In this paper, we give a generalization of it. In our case, the Frobenius matrix is no longer described by p-adic gamma function, and then we describe it by the p-adic polygamma functions. Since the p-adic polygamma values are linear combinations of p-adic L-values of Dirichlet characters, it turns out that the Frobenius matrix is described by p-adic L-values. Our result has an application to the study on Frobenius on p-adic cohomology. We show that, for a projective smooth family such that the Picard-Fuchs equation is a hypergeometric equation, the Frobenius matrix on the log-crystalline cohomology is described by some values of the logarithmic function and p-adic L-functions of Dirichlet characters. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_08940 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Frobenius structure on hypergeometric equations, p-adic polygamma values and p-adic L-values Asakura, Masanori Hagihara, Kei Number Theory 12H25 (primary), 11S80 (secondary) Recently, Kedlaya proves certain formula describing explicitly the Frobenius structure on a hypergeometric equation. In this paper, we give a generalization of it. In our case, the Frobenius matrix is no longer described by p-adic gamma function, and then we describe it by the p-adic polygamma functions. Since the p-adic polygamma values are linear combinations of p-adic L-values of Dirichlet characters, it turns out that the Frobenius matrix is described by p-adic L-values. Our result has an application to the study on Frobenius on p-adic cohomology. We show that, for a projective smooth family such that the Picard-Fuchs equation is a hypergeometric equation, the Frobenius matrix on the log-crystalline cohomology is described by some values of the logarithmic function and p-adic L-functions of Dirichlet characters. |
| title | Frobenius structure on hypergeometric equations, p-adic polygamma values and p-adic L-values |
| topic | Number Theory 12H25 (primary), 11S80 (secondary) |
| url | https://arxiv.org/abs/2307.08940 |