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?

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    $\begingroup$ I think it is conjectured that the answer is yes, but this is very far from being proven. $\endgroup$ – 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 $n=2$.

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  • $\begingroup$ Thank you for your detailed response, that's what I expected, I just wanted to be sure. $\endgroup$ – user130124 Oct 15 '18 at 14:53

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