Automorphic classification of different types of abelian surfaces

Question

For elliptic curves over $\mathbb{Q}$ the Mumford-Tate group is either $\mathrm{GL}_2$ or $\mathrm{Res}_\mathbb{Q}^F (\mathbb{G}_m)$ if it has CM with the imaginary quadratic field $F$. In this case the $\mathbb{Q}$-endomorphism algebra completely determines the Mumford-Tate group. On the automorphic side, having CM can be seen as the associated modular form $f$ satisfying $f=f\otimes \chi$ for some quadratic character $\chi$.

For a simple abelian surface $A/\mathbb{Q}$, again the endomorphism algebra determines the Mumford-Tate group. Here one can find the classification of the Hodge group (special Mumford-Tate group) for low dimensional abelian varieties. In particular there are four cases for abelian surfaces. One expects to be able to associate an automorphic representation $\pi$ of $\mathrm{GSp}_4$ to the abelian surface $A$.

My question is that if the four cases of Mumford-Tate groups (or the endomorphism algebra if you like) can be translated to properties (hopefully some kind of symmetries) of the automorphic representation $\pi$. For instance, I think the CM case (Type IV(2,1) here) should be similar to the case of elliptic curves, namely $\pi=\pi \otimes \chi$ for some character.

My main concern is the two dimensional case but any comments on the higher dimensional cases where the Mumford-Tate group is not determined by the endomorphism algebra is also highly appreciated.

Answer 1

Yes, knowing the endomorphism algebra of $A$ (conjecturally) translates to certain properties of an associated automorphic representation $\pi$.

First, you should look at the Galois type, which is labelled (A)-(F) in FKRS's paper on Sato-Tate groups. (These labels also include non-simple $A$, which correspond to products of elliptic curves, and thus products of rational elliptic newforms.)

If the $\mathbb Q$-endomorphism algebra contains a real quadratic field $K$, i.e., $A$ has real multiplication, then $A$ is of GL(2) type as defined by Ribet. By Ribet + Serre's conjectures, this means $A$ corresponds to weight 2 newform of GL(2) with rationality field $K$. If further $A$ has quartic CM, then $f$ will have CM, and have inner twists.

Otherwise, then the (generalized) paramodular conjecture predicts $A$ corresponds to a weight 2 automorphic representation $\pi$ of GSp(4). See Brumer and Kramer's formulation and/or BCGP's potential modularity paper. In the typical case (trivial endomorphisms), the Galois representations are strongly irreducible, but the other Galois types (B)-(F) correspond to representations that are reducible upon restriction to some finite extension $K/\mathbb Q$, with details depending on the type---see Section 9.2 of [BCGP]. Consequently this means that $\pi_K$ is an isobaric sum, which is one way of stating the automorphic property corresponding to the endomorphism algebra of $A$.

There should be similar stories in higher dimensions. Certainly the GL(2)-type situation is understood in higher dimensions.