On the motive of an abelian scheme with non-trivial endomorphisms

Details On the motive of an abelian scheme with non-trivial endomorphisms On the motive of an abelian scheme with non-trivial endomorphisms

2011-10-19

arxiv.org/abs/1110.4264v1

Mathematics

Algebraic Geometry

13 pages

Scientific paper

Let X be an abelian scheme over a base variety S and let D = End(X/S) \otimes Q be its endomorphism algebra. We prove that the relative Chow motive of X has a natural decomposition as a direct sum of motives R^(alpha) where alpha runs over an explicitly determined finite set. To each alpha corresponds an irreducible representation rho_alpha of the group D^{opp,*} and the motivic decomposition is such that R^(alpha), as a functor on the category of relative Chow motives, is a sum of copies of rho_alpha. In particular the Chow group CH(R^(alpha)), as a representation of D^{opp,*}, is a sum of copies of rho_alpha. Our decomposition refines the motivic decomposition of Deninger and Murre, as well as Beauville's decomposition of the Chow group.

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