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YCor
YCor
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Are there GalrepsGalois representations associated with any RACregular algebraic cuspidal automorphic representation?

Let $K$ be a number field and $\pi$ be a regular algebraic cuspidal automorphic representation on $GL_2(\mathbb{A}_K)$$\mathrm{GL}_2(\mathbb{A}_K)$. Let $\lambda$ be a prime of the field of Fourier coefficients of $\pi$ and $\mathcal{O}$ the completion of the ring of integers of the field of Fourier coefficients of $\pi$. Does there always exist a Galois representation $\rho_{\pi,\lambda}:Gal(\bar{K}/K) \rightarrow GL_2(\mathcal{O})$$\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 Galreps associated with any RAC automorphic representation?

Let $K$ be a number field and $\pi$ be a regular algebraic cuspidal automorphic representation on $GL_2(\mathbb{A}_K)$. 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 $\rho_{\pi,\lambda}:Gal(\bar{K}/K) \rightarrow GL_2(\mathcal{O})$ with no hypothesis on the field $K$, or must it be CM?

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

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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user130124
user130124

Are there Galreps associated with any RAC automorphic representation?

Let $K$ be a number field and $\pi$ be a regular algebraic cuspidal automorphic representation on $GL_2(\mathbb{A}_K)$. 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 $\rho_{\pi,\lambda}:Gal(\bar{K}/K) \rightarrow GL_2(\mathcal{O})$ with no hypothesis on the field $K$, or must it be CM?