Is the Sato-Tate conjecture known for Bianchi modular forms?

Question

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)

Answer 1

Does it follow from the work of [1] that the Sato-Tate conjecture is known for some class of cuspidal automorphic forms for GL2 over CM fields?

Tautologically: "yes, those which correspond to modular elliptic curves". These are a (strict) subset of the forms which are cohomological of trivial weight (i.e. parallel weight 2 in your terminology) and have trivial character and Hecke eigenvalues in $\mathbf{Q}$.

More relevantly: it looks to me as if their argument works for any newform of trivial weight, whatever the character and the coefficient field. These should still give rise to "very weakly compatible systems" as in Corollary 7.1.12 of op.cit., of which the Sato--Tate result is a special case.

The restriction on the weight is more serious; one would certainly expect something of this sort to be true for arbitrary cohomological weights, but the ten-author paper does not cover this case, since they require that their Galois representations have all Hodge--Tate weights 0 or 1. It would take a genuine expert [not me!] to say how easy it would be to extend their results to cover this case.

What is the precise formulation of the Sato-Tate conjecture for general cuspidal automorphic forms for GL2 over CM fields?

For parallel weight and trivial character, the statement is exactly what you think it is. For more general (cohomological) weights, there are some fiddly but ultimately minor complications of book-keeping (which are already present over totally real fields too): you have to be careful about how to normalise $c(\mathfrak{p})$, with various conventions differing by integer or half-integer powers of $\mathrm{Nm}(\mathfrak{p})$. The "analytic" normalisation is such that the L-series $\sum c(\mathfrak{n}) \mathrm{Nm}(\mathfrak{n})^{-s}$ has its functional equation centred at $s = \tfrac{1}{2}$; then the Ramanujan conj. is that $|c(\mathfrak{p})| \le 2$ for all $\mathfrak{p}\nmid \mathfrak{N}$ where $\mathfrak{N}$ is the level. (Note that equality can occur, even in the parallel weight 2 case, although it occurs with "probability 0" in some sense.)

From here, it should be clear how to state the Sato--Tate conj if $\pi$ has trivial character, since then $c(\mathfrak{p})$ is real, and we're just asserting that its distribution in $[-2, 2]$ is the expected one. To state the Sato–Tate conjecture for general characters $\chi$, one can argue as follows: $\chi$ will factor through the ray class group of level $\mathfrak{N}$; for each $x \in Cl(\mathfrak{N}) / \ker(\chi)$, choose some $\alpha_x$ such that $\alpha^2 = \chi(x)$. Then $c'(\mathfrak{p}) = c(\mathfrak{p}) / 2\alpha_{[\mathfrak{p}]}$ will be in the real interval $[-1, 1]$, and Sato--Tate predicts that the ratios $c'(\mathfrak{p})$ have the expected distribution.

Answer 2

Let $K/\mathbb{Q}$ be CM extension. Suppose that a primitive Bianchi cusp form $F$ corresponds with a cuspidal automorphic representation of $\mathrm{GL}_2(\mathbb{A}_K)$ that is not the automorphic induction from an algebraic Hecke character defined over a quadratic extension of $K$. Then the Sato--Tate conjecture (along with the generalized Ramanujan conjecture) is now a theorem for $F$. This was recently proved by Boxer, Calegari, Gee, Newton, and Thorne (https://arxiv.org/pdf/2309.15880.pdf). See Theorems A and B therein.