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You can find answers to your questions in: Deligne, Pierre: Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques. Automorphic forms, representations and L …

The answer is Yes. We denote $K(G)=G({\mathbb Q})_+/\rho G^{\rm sc}({\mathbb Q})$. We compute $K(G)$; see the corollary below. It is clear from the corollary that $K(G)$ is canonically isomorphic to $ …

Consider the cyclotomic field $L={{\mathbb{Q}}}(\zeta_8)={{\mathbb{Q}}}(\sqrt{2},i)$, where $\zeta_8$ is a primitive 8-th root of unity. Let $\Lambda={{\mathbb{Z}}}[\zeta_8]$ denote the ring of integ …

This question is related to my previous question, to which I got a partial answer. Consider the cyclotomic field $L={{\mathbb{Q}}}(\zeta_8)={{\mathbb{Q}}}(\sqrt{2},i)$, where $\zeta_8$ is a primitive …

No. Here is a counter-example (I add details to my comment). Let $F$ be a totally real number field. Consider the group $G_0=R_{F/\mathbb{Q}}\mathrm{SL}_{2,F}$, then $G_0$ naturally embeds into $\ma …

The answer depends on your definition of a Shimura pair $(G,X)$. Look in Section 2.1 of Deligne's paper. If you assume only axioms (2.1.1.1), (2.1.1.2) and (2.1.1.3), then any number field $F$ can o …

First question: No, these conditions are not sufficient. For details see: Deligne, Pierre, Variétés de Shimura: interprétation modulaire, et techniques de construction de modèles canoniques. (French …

The question is essentially about ${\mathbf{R}}$-groups, so we shall assume that $G$ is defined over $\mathbf{R}$. It is not true that for a given connected reductive ${\mathbf{R}}$-group $G$, there …