Tate module tensor decompositions and the Sato–Tate conjecture for certain abelian varieties potentially of $$\mathrm {GL}_2$$ GL 2 -type

We introduce a tensor decomposition of the $$\ell $$ -adic Tate module of an abelian variety $$A_0$$ defined over a number field which is geometrically isotypic. If $$A_0$$ is potentially of $$\mathrm {GL}_2$$ -type and defined over a totally real number field, we use this decomposition to describe its Sato–Tate group and to prove the Sato–Tate conjecture in certain cases.