Originally formulated for elliptic curves, the Sato-Tate conjecture regarding the equidistribution of Frobenius trace values according to the Haar measure on a certain compact group (the Sato-Tate group) was generalised in 1994 by Serre to any motive over a number field - see 13.5? in [5]. As is now well known, the Sato-Tate conjecture has been proven for elliptic curves over totally real fields [3] as well as for all Hilbert modular forms [2].

More recently, in 2018, the authors in [1] prove modularity lifting theorems for Galois representations over CM fields which yield several attractive corollaries including the Sato-Tate conjecture for elliptic curves over CM fields.

My main question is the following:

### Does it follow from the work of [1] that the Sato-Tate conjecture is known for some class of cuspidal automorphic forms for $\mbox{GL}_2$ over CM fields?

I am chiefly concerned with the case of the CM field being an imaginary quadratic field, in which case the automorphic forms are often called *Bianchi modular forms*, hence the question in the title. A concise account of cusp forms for $\mbox{GL}_2$ associated to more general CM fields may be found in Section 2 of [4].

If I may hazard a guess at my question above, I would think that it is currently known only for automorphic $\mbox{GL}_2$ newforms over CM fields of parallel weight 2 with rational Fourier coefficients. But I am not an expert in this area, hence my question on this forum.

I am also curious to know the following:

### What is the precise formulation of the Sato-Tate conjecture for general cuspidal automorphic forms for $\mbox{GL}_2$ over CM fields?

I think [1] also establishes the Generalised Ramanujan Conjecture (GRC) in the parallel weight 2 setting. For general parallel weight $k$, does GRC imply that there is an inequality

$$ \left|\frac{c(\frak{p})}{\mbox{Nm}({\frak{p}})^{\frac{k-1}{2}}}\right| < 2, $$

where the $c(\frak{p})$ are the Fourier coefficients of the (cuspidal new $\mbox{GL}_2$) automorphic form at the integral prime ideals of the CM field $K$? If so, then I can begin to see how one can at least start talking about equidistribution.

References:

[1]: P. B. Allen, F. Calegari, A. Caraiani, T. Gee, D. Helm, B. V. Le Hung, J. Newton, P. Scholze, R. Taylor, J.A. Thorne. *Potential automorphy over CM fields*. arXiv:1812.09999 (2018)

[2]: T.Barnet-Lamb, T. Gee, D.Geraghty. *The Sato-Tate cojecture for Hilbert Modular forms*. JAMS (2011)

[3]: T. Barnet-Lamb, D. Geraghty, M. Harris and R. Taylor . *A family of Calabi-Yau varieties and potential automorphy II*. P.R.I.M.S. 47 (2011)

[4]: E. Ghate. *Critical values of twisted tensor L-functions over CM-Fields*. Proc. Symp. Pure Math. (1999)

[5]: J.P. Serre. *Propriétés conjecturales des groupes de Galois motiviques et des représentations $\ell$-adiques*. Proc. Symp. Pure Math. (1994)

for GL2over CM fields"? Automorphic forms make sense on any reductive group, but I don't think much is known about Sato--Tate for anything beyond GL2, except a few partial results for things like GSp4. $\endgroup$