  For a certain class of vector bundles E on abelian varieties A over local
fields containing all line bundles algebraically equivalent to zero we define a
canonical representation of the Tate module of A on the fibre of E in the zero
section. This extends an old construction of Tate for line bundles to vector
bundles of higher rank. We also compare this construction to the theory of
parallel transport for vector bundles on p-adic curves developed in
mathAG/0403516. Relations with the Hodge-Tate decomposition are also explained.
