Decomposition de motifs abeliens

TL;DR

The paper provides an explicit algebraic description of certain endomorphisms related to abelian varieties and demonstrates how this leads to lifting the canonical construction functor to the category of Chow motives for Shimura varieties of PEL type.

Contribution

It explicitly characterizes a $Q$-algebra of correspondences for abelian varieties and extends the canonical construction to Chow motives for PEL Shimura varieties.

Findings

Explicit presentation of a $Q$-algebra of correspondences

Isomorphism between algebra of correspondences and endomorphisms respecting Lefschetz group

Lifting of the canonical construction functor to Chow motives

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.