Décomposition de motifs abéliens

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 $cl_{|_{B_{i,r}}}: B_{i,r} \otimes_{\mathbb{Q}} F \cong End_{Lef(A)}(H^i(A^r,F)).$ 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} ${\mu: Rep (G) \rightarrow VHS(S(\mathbb{C}))}$ lifts to a functor ${\tilde{\mu}: Rep (G) \rightarrow CHM(S)_{\mathbb{Q}}}$, where $CHM(S)_{\mathbb{Q}}$ is the category of relative Chow motives.