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Height Pairings of 1-Motives

Abstract : The purpose of this work is to generalize, in the context of 1-motives, the height pairings constructed by B. Mazur and J. Tate on abelian varieties. Following their approach, we consider ρ-splittings of the Poincaré biextension of a 1-motive and require that they be compatible with the canonical linearization associated to the biextension. We establish results concerning the existence of such ρ-splittings. When ρ is unramified this is guaranteed if the monodromy pairing of the 1-motive considered is non-degenerate. For ramified ρ, the ρ-splitting is constructed from a pair of splittings of the Hodge filtrations of the de Rham realizations of the 1-motive and its dual. This generalizes previous results by R. Coleman and Y. Zarhin for abelian varieties. These ρ-splittings are then used to define a global pairing between rational points of a 1-motive and its dual. We also provide local pairings between zero cycles and divisors on a variety, which is done by applying the previous results to its Picard and Albanese 1-motives.
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Submitted on : Tuesday, November 27, 2018 - 3:04:10 PM
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Carolina Rivera Arredondo. Height Pairings of 1-Motives. General Mathematics [math.GM]. Université de Bordeaux; Università degli studi (Milan, Italie), 2018. English. ⟨NNT : 2018BORD0082⟩. ⟨tel-01936646⟩

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