Algebraic independence of the Carlitz period and its hyperderivatives
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arXiv
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| Format: | Preprint |
| Publié: |
2021
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| _version_ | 1866908844605571072 |
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| author | Maurischat, Andreas |
| author_facet | Maurischat, Andreas |
| contents | This paper deals with the fundamental period $\tildeπ$ of the Carlitz module. The main theorem states that the Carlitz period and all its hyperderivatives are algebraically independent over the base field $\mathbb{F}_q(θ)$. Our approach also reveals a connection of these hyperderivatives with the coordinates of a period lattice generator of the tensor powers of the Carlitz module which was already observed by M. Papanikolas in a yet unpublished paper. Namely, these coordinates can be obtained by explicit polynomial expressions in $\tildeπ$ and its hyperderivatives. Papanikolas also gave various presentations of these expressions which we also prove here. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2104_02630 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Algebraic independence of the Carlitz period and its hyperderivatives Maurischat, Andreas Number Theory 11J93 This paper deals with the fundamental period $\tildeπ$ of the Carlitz module. The main theorem states that the Carlitz period and all its hyperderivatives are algebraically independent over the base field $\mathbb{F}_q(θ)$. Our approach also reveals a connection of these hyperderivatives with the coordinates of a period lattice generator of the tensor powers of the Carlitz module which was already observed by M. Papanikolas in a yet unpublished paper. Namely, these coordinates can be obtained by explicit polynomial expressions in $\tildeπ$ and its hyperderivatives. Papanikolas also gave various presentations of these expressions which we also prove here. |
| title | Algebraic independence of the Carlitz period and its hyperderivatives |
| topic | Number Theory 11J93 |
| url | https://arxiv.org/abs/2104.02630 |