On Galois representations for abelian varieties with complex and real multiplications. (Q1394912)

This paper studies the image of \(\ell\)-adic representations coming from the Tate module of an abelian variety. Let \(A\) be a simple abelian variety of nondegerate CM-type defined over a number field \(F\) with \(\text{End}(A)\otimes\mathbb{Q}= E\subset F\), and let \(\phi:T_{E'}\to T_E\) be the homomorphism of tori defined in [\textit{K. A. Ribet}, Mém. Soc. Math. Fr., Nouv. Sér. 2, 75--94 (1980; Zbl 0452.14009)], where \(E'\) denotes the reflex field of \(E\). The authors determine the image of \(\overline{\rho_\ell}: G_F\to \text{GL}_{2g}(\mathbb{F}_\ell)\) explicitly under the assumption that the kernel of the map \(\phi\) is connected. When \(\text{End}(A)\otimes\mathbb{Q}\) is a totally real field of degree \(e\) with \(g/e\) odd, they determine the commutator subgroup of \(\overline{\rho_\ell}(G_F)\) under some conditions on \(\ell\). As a consequence they show the validity of the Mumford-Tate conjecture for such \(A\).