Decomposition de motifs abeliens

Abstract

Let A be an abelian variety and let us fix a Weil cohomology with coefficients in F. Let $H^1(A,F)$ be the first cohomology group of A and $Lef(A)\subset GL(H^1(A,F))$ be its Lefschetz group, i.e. the sub-group of $GL(H^1(A,F))$ of linear applications commuting with endomorphisms of A and respecting the pairing induced by a polarization. We give an explicit presentation of a $\mathbb{Q}$-algebra of correspondences $B_{i,r}$ such that the cycle class map induces an isomorphism $c{l}_{{|}_{B_{i,r}}}:B_{i,r}{\otimes }_{\mathbb{Q}}F\cong En{d}_{Lef(A)}(H^i(A^r,F)).$ We also give relative versions of this result. We deduce in particular the following fact. Let $S=S_K(G,\mathcal{X})$ be a Shimura variety of PEL type. Then the functor \textit{canonical construction} $\mu :Rep(G)\to VHS(S(\mathbb{C}))$ lifts to a functor $\tilde{\mu }:Rep(G)\to CHM(S{)}_{\mathbb{Q}}$, where $CHM(S{)}_{\mathbb{Q}}$ is the category of relative Chow motives.

Final version. Added application to PEL Shimura varieties. 22 pages, in french, Manuscripta Math., 2014

References in corpus (2)

• The boundary motive: definition and basic properties
• Degeneration of Hodge structures over Picard modular surfaces

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