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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2018
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| Acceso en línea: | https://arxiv.org/abs/1812.11576 |
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| _version_ | 1866908338324766720 |
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| author | Grishkov, A. Logachev, D. |
| author_facet | Grishkov, A. Logachev, D. |
| contents | Let $M$ be an uniformizable Anderson $t$-motive of rank $r$, $L$ its lattice and $l_*:=\{l_1,\dots, l_r\}$ its basis. We define a map $δ$ from the set of these bases to a flag variety (the present text gives the definition of $δ$ only for elements of the maximal Schubert cell, and for few other cases). If $l_*$ belongs to the maximal Schubert cell then $δ(l_*)$ is described as a set of matrices parametrized by integer points of a tetrahedron; they are called the Siegel element of $l_*$. We give explicit formulas for a Siegel element for $M'$ -- the dual of $M$. As a by-product, we get another proof of the theorem that the lattice of $M'$ is the dual of the lattice of $M$, independent of the proof obtained by U. Hartl and A.-K. Juschka. Generalizations of this result to non-maximal Schubert cells, other tensor operations (tensor product and Hom) and t-motives having non-trivial endomorphism rings, are subjects of further research. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1812_11576 |
| institution | arXiv |
| publishDate | 2018 |
| record_format | arxiv |
| spellingShingle | Lattice of the dual of an Anderson $t$-motive in terms of a map to a flag variety Grishkov, A. Logachev, D. Number Theory 11G09, 14M15 Let $M$ be an uniformizable Anderson $t$-motive of rank $r$, $L$ its lattice and $l_*:=\{l_1,\dots, l_r\}$ its basis. We define a map $δ$ from the set of these bases to a flag variety (the present text gives the definition of $δ$ only for elements of the maximal Schubert cell, and for few other cases). If $l_*$ belongs to the maximal Schubert cell then $δ(l_*)$ is described as a set of matrices parametrized by integer points of a tetrahedron; they are called the Siegel element of $l_*$. We give explicit formulas for a Siegel element for $M'$ -- the dual of $M$. As a by-product, we get another proof of the theorem that the lattice of $M'$ is the dual of the lattice of $M$, independent of the proof obtained by U. Hartl and A.-K. Juschka. Generalizations of this result to non-maximal Schubert cells, other tensor operations (tensor product and Hom) and t-motives having non-trivial endomorphism rings, are subjects of further research. |
| title | Lattice of the dual of an Anderson $t$-motive in terms of a map to a flag variety |
| topic | Number Theory 11G09, 14M15 |
| url | https://arxiv.org/abs/1812.11576 |