# Automorphic representations attached to abelian varieties

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 L-function agrees with the L-function 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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Are you using Langlands functoriality to lift these automorphic representations? I understand how to pass from abelian variety to $Sp(2d)$ to $GL(2d)$, but what is your connection with orthogonal group? (Liberally insert conjectural(ly) throughout.) – Victor Protsak Jul 13 '10 at 0:18
The Galois representation attached to $T_\ell(A)$ has image in $Sp_{2d}(\overline{\mathbb{Q}_{\ell}})$, and $Sp_{2d}$ is the dual group of $SO(2d+1)$. – David Hansen Jul 13 '10 at 1:51
(whoops) - so, yes, I am using functoriality. :) – David Hansen Jul 13 '10 at 1:54
Don't you intend to use automorphic forms on something whose dual group is GSp(2d)? I think one would usually write GSpin(2d + 1)/Q for the split group over Q with the appropriate root datum. Since the Galois representations associated to you abelian variety A/Q are representations (and not cocycles), one would expect an automorphic form on GSpin(2d + 1)/Q (and not an outer form). I don't know whether one would expect to be able to transfer to some inner form G/Q that is not split at R, but perhaps you are not worried about that. – user2490 Jul 13 '10 at 1:57
I think that you may want automorphic forms on GSpin(2d+1)/Q (split with root datum dual to that of GSp(2d)) and not on Spin(2d+1)/Q (split with root datum dual to that of PSp(2d)), unless you are projectivizing your Galois representations (into PSp(2d) instead of GSp(2d)). I don't follow your remark about the multiplicity of spin groups. The most general place to imagine your automorphic forms is on the (unique) split reductive group over Q with the described root datum. You may be able to transfer to an inner form over Q, depending on ramification. – user2490 Jul 13 '10 at 14:11