The aim of this paper is to investigate the Galois module structure of the Tate module of an abelian variety defined over a number field. We focus on simple abelian varieties of type IV in Albert classification. We describe explicitly the decomposition of the O_λ[G_F]-module T_λ(A) into components that are rationally and residually irreducible. Moreover these components are non-degenerate, hermitian modules that rationally and residually are non-degenerate, hermitian vector spaces.

The authors would like to thank the referee for valuable comments, suggestions and corrections which improved the exposition of the paper.