Line bundles and p-adic characters
  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. 
Christopher Deninger, Annette Werner