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Let $G_{\mathbb{Q}}$ be the absolute Galois group of the rationals, and let $F: \mathrm{Aff}/\textbf{Q}_p\longrightarrow \mathrm{Sets}$ be the functor which associates to every affine $\mathbb{Q}_p$ scheme $\operatorname{Spec} A$ the set of representations $\operatorname{Hom}(G_{\mathbb{Q}}, \operatorname{GL}_2(A))$.

Is this functor representable?

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  • $\begingroup$ Representable by a scheme? This seems hopeless since the pushout of rings need not induce a pushout of GL2 in groups. $\endgroup$
    – Kenta Suzuki
    Commented Jun 16, 2023 at 20:47

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