Let $A$ be an abelian variety defined over $\mathbb{Q}$, of dimension $d$. It is widely expected that there is an automorphic representation $\pi_A$ of $GL(2d)/\mathbb{Q}$ whose Lfunction agrees with the Lfunction attached to the Tate module of $A$. In fact, $\pi_A$ should arise from an automorphic form on an orthogonal group. My question is, which (real) form of $O(2d+1)$ should $\pi_A$ live on? For $d=1$ the group is $O(2,1)$, which is isogenous to $SL_2$, and for $d=2$ the group should be is $O(2,3)$, which is isogenous to $Sp_4$. But what should happen in general?
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