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| Format: | Preprint |
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2019
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| Online Access: | https://arxiv.org/abs/1910.11057 |
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| _version_ | 1866912338817318912 |
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| author | Mornev, M. |
| author_facet | Mornev, M. |
| contents | A Drinfeld module has a $\mathfrak{p}$-adic Tate module not only for every finite place $\mathfrak{p}$ of the coefficient ring but also for $\mathfrak{p} = \infty$. This was discovered by J.-K. Yu in the form of a representation of the Weil group. Following an insight of Taelman we construct the $\infty$-adic Tate module by means of the theory of isocrystals. This applies more generally to pure $A$-motives and to pure $F$-isocrystals of $p$-adic cohomology theory. We demonstrate that a Drinfeld module has good reduction if and only if its $\infty$-adic Tate module is unramified. The key to the proof is the theory of Hartl and Pink which gives an analytic classification of vector bundles on the Fargues-Fontaine curve in equal characteristic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1910_11057 |
| institution | arXiv |
| publishDate | 2019 |
| record_format | arxiv |
| spellingShingle | Tate modules of isocrystals and good reduction of Drinfeld modules Mornev, M. Number Theory A Drinfeld module has a $\mathfrak{p}$-adic Tate module not only for every finite place $\mathfrak{p}$ of the coefficient ring but also for $\mathfrak{p} = \infty$. This was discovered by J.-K. Yu in the form of a representation of the Weil group. Following an insight of Taelman we construct the $\infty$-adic Tate module by means of the theory of isocrystals. This applies more generally to pure $A$-motives and to pure $F$-isocrystals of $p$-adic cohomology theory. We demonstrate that a Drinfeld module has good reduction if and only if its $\infty$-adic Tate module is unramified. The key to the proof is the theory of Hartl and Pink which gives an analytic classification of vector bundles on the Fargues-Fontaine curve in equal characteristic. |
| title | Tate modules of isocrystals and good reduction of Drinfeld modules |
| topic | Number Theory |
| url | https://arxiv.org/abs/1910.11057 |