# Are there Galois representations associated with any regular algebraic cuspidal automorphic representation?

Let $$K$$ be a number field and $$\pi$$ be a regular algebraic cuspidal automorphic representation on $$\mathrm{GL}_2(\mathbb{A}_K)$$. Let $$\lambda$$ be a prime of the field of Fourier coefficients of $$\pi$$ and $$\mathcal{O}$$ the completion of the ring of integers of the field of Fourier coefficients of $$\pi$$. Does there always exist a Galois representation $$\rho_{\pi,\lambda}:\mathrm{Gal}(\bar{K}/K) \rightarrow \mathrm{GL}_2(\mathcal{O})$$ with no hypothesis on the field $$K$$, or must it be CM?

• I think it is conjectured that the answer is yes, but this is very far from being proven. – GH from MO Oct 15 '18 at 0:26

In Motifs et formes automorphes: applications du principe de fonctorialité by Laurent Clozel (in Automorphic forms, Shimura varieties and $$L$$-functions Volume I (1990)), it is asked in 4.3.2 whether the category of algebraic automorphic representation of $$\operatorname{GL}_n(\mathbb A_F)$$ defined over $$\bar{\mathbb Q}$$ could be a tannakian category equivalent to the category of motives over $$F$$ (with coefficients in $$\bar{\mathbb Q}$$). This entails (Conjecture 4.5 of loc. cit.) that there is even a motivic Galois representation attached to any cuspidal algebraic (isobaric) automorphic representation of $$\operatorname{GL}_n(\mathbb A_F)$$. But as GH says, this is far from known, even when $$n=2$$.