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?



Are there Galois representations associated with any regular algebraic cuspidal automorphic representation?
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 $n=2$.


