LOCAL INDECOMPOSABILITY OF TATE MODULES OF NON-CM ABELIAN VARIETIES WITH REAL MULTIPLICATION.

The article discusses the indecomposability of p-adic Galois representations emerging from the Tate module of an abelian variety with real multiplication defined over a number field. The indecomposability for the nearly ordinary Galois representation of each weight of Hilbert cusp form is mentioned. The importance of the p-local indecomposability of the p-adic Tate module as a characterization of non-CM/CM abelian varieties is also addressed.