Mumford-Tate groups of 1-motives and Weil pairing
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866910588767043584 |
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| author | Bertolin, Cristiana Philippon, Patrice |
| author_facet | Bertolin, Cristiana Philippon, Patrice |
| contents | We show how the geometry of a 1-motive $M$ (that is existence of endomorphisms and relations between the points defining it) determines the dimension of its motivic Galois group ${\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}(M)$. Fixing periods matrices $Π_M$ and $Π_{M^*}$ associated respectively to a 1-motive $M$ and to its Cartier dual $M^*,$ we describe the action of the Mumford-Tate group of $M$ on these matrices. In the semi-elliptic case, according to the geometry of $M$ we classify polynomial relations between the periods of $M$ and we compute exhaustively the matrices representing the Mumford-Tate group of $M$. This representation brings new light on Grothendieck periods conjecture in the case of 1-motives. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_16301 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Mumford-Tate groups of 1-motives and Weil pairing Bertolin, Cristiana Philippon, Patrice Algebraic Geometry Number Theory We show how the geometry of a 1-motive $M$ (that is existence of endomorphisms and relations between the points defining it) determines the dimension of its motivic Galois group ${\mathcal{G}}{\mathrm{al}}_{\mathrm{mot}}(M)$. Fixing periods matrices $Π_M$ and $Π_{M^*}$ associated respectively to a 1-motive $M$ and to its Cartier dual $M^*,$ we describe the action of the Mumford-Tate group of $M$ on these matrices. In the semi-elliptic case, according to the geometry of $M$ we classify polynomial relations between the periods of $M$ and we compute exhaustively the matrices representing the Mumford-Tate group of $M$. This representation brings new light on Grothendieck periods conjecture in the case of 1-motives. |
| title | Mumford-Tate groups of 1-motives and Weil pairing |
| topic | Algebraic Geometry Number Theory |
| url | https://arxiv.org/abs/2210.16301 |