Representations and factors of restriction of scalars

Let L/K be a Galois extension of number fields and let A be an abelian variety defined over K. In this paper we establish the relation between the irreducible characters of the Galois group Gal(L/K) and the simple factors of the restriction of scalars ResL/K(A) of A from L to K. Then we derive some equivalences of Birch and Swinnerton–Dyer conjectures.