$\mathrm{GL}_2$-representations with maximal image

For a matrix group $\mathcal{G}$, consider a Galois representation\[\varphi : \mathrm{Gal}(\overline{\mathbb{Q}} / \mathbb{Q}) \longrightarrow \mathcal{G}(\hat{\mathbb{Z}})\]which extends the cyclotomic character. For a broad class of matrix groups $\mathcal{G}$, we prove a theorem characterizing when such a representation has image which is “as large as possible” inside a fixed open subgroup $G \subseteq \mathcal{G}(\hat{\mathbb{Z}})$. As applications, we obtain such a characterization for the Galois representation on the torsion of a simple principally polarized $k$-dimensional abelian variety A defined over $\mathbb{Q}$ (where $\mathcal{G} = \mathrm{GSp}_{2k}$) and also for the Galois representation on the torsion of a product of $k$ elliptic curves over $\mathbb{Q}$ (where $\mathcal{G} = \lbrace (g_1, \dotsc , g_k) \in \mathrm{GL}^k_2 : \det g_1 = \cdots = \det g_k \rbrace$). Our results are motivated by open image theorems for classes of abelian varieties initiated by Serre in the 1960s.

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Published 20 May 2015