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Article Dans Une Revue Journal of Pure and Applied Algebra Année : 2024

Mumford-Tate groups of 1-motives and Weil pairing

Cristiana Bertolin
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Résumé

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 $\Pi_M$ and $\Pi_{M^*}$ 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.
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Dates et versions

hal-04594962 , version 1 (30-05-2024)

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Cristiana Bertolin, Patrice Philippon. Mumford-Tate groups of 1-motives and Weil pairing. Journal of Pure and Applied Algebra, 2024, 228 (10), pp.107702. ⟨10.1016/j.jpaa.2024.107702⟩. ⟨hal-04594962⟩
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