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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2409.04029 |
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| _version_ | 1866912323252256768 |
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| author | Kędzierski, Dawid E. Krasoń, Piotr |
| author_facet | Kędzierski, Dawid E. Krasoń, Piotr |
| contents | In the work of M. A. Papanikolas and N. Ramachandran [A Weil-Barsotti formula for Drinfeld modules, Journal of Number Theory 98, (2003), 407-431] the Weil-Barsotti formula for the function field case concerning $\Ext_τ^1(E,C)$ where $E$ is a Drinfeld module and $C$ is the Carlitz module was proved. We generalize this formula to the case where $E$ is a strictly pure \tm module $Φ$ with the zero nilpotent matrix $N_Φ.$ For such a \tm module $Φ$ we explicitly compute its dual \tm module $Φ^{\vee}$ as well as its double dual $Φ^{{\vee}{\vee}}.$ This computation is done in a a subtle way by combination of the \tm reduction algorithm developed by F. Głoch, D.E. K{\k e}dzierski, P. Kraso{ń} [ Algorithms for determination of \tm module structures on some extension groups , arXiv:2408.08207] and the methods of the work of D.E. K{\k e}dzierski and P. Kraso{ń} [On $\Ext^1$ for Drinfeld modules, Journal of Number Theory 256 (2024) 97-135]. We also give a counterexample to the Weil-Barsotti formula if the nilpotent matrix $N_Φ$ is non-zero. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_04029 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Weil-Barsotti formula for $\mathbf{T}$-modules Kędzierski, Dawid E. Krasoń, Piotr Number Theory In the work of M. A. Papanikolas and N. Ramachandran [A Weil-Barsotti formula for Drinfeld modules, Journal of Number Theory 98, (2003), 407-431] the Weil-Barsotti formula for the function field case concerning $\Ext_τ^1(E,C)$ where $E$ is a Drinfeld module and $C$ is the Carlitz module was proved. We generalize this formula to the case where $E$ is a strictly pure \tm module $Φ$ with the zero nilpotent matrix $N_Φ.$ For such a \tm module $Φ$ we explicitly compute its dual \tm module $Φ^{\vee}$ as well as its double dual $Φ^{{\vee}{\vee}}.$ This computation is done in a a subtle way by combination of the \tm reduction algorithm developed by F. Głoch, D.E. K{\k e}dzierski, P. Kraso{ń} [ Algorithms for determination of \tm module structures on some extension groups , arXiv:2408.08207] and the methods of the work of D.E. K{\k e}dzierski and P. Kraso{ń} [On $\Ext^1$ for Drinfeld modules, Journal of Number Theory 256 (2024) 97-135]. We also give a counterexample to the Weil-Barsotti formula if the nilpotent matrix $N_Φ$ is non-zero. |
| title | Weil-Barsotti formula for $\mathbf{T}$-modules |
| topic | Number Theory |
| url | https://arxiv.org/abs/2409.04029 |