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On Galois representations for abelian varieties with complex and real multiplications. (English) Zbl 1056.11034
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 [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 \(A\).

MSC:
11G10 Abelian varieties of dimension \(> 1\)
11G15 Complex multiplication and moduli of abelian varieties
11R34 Galois cohomology
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