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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 …

@Brian Conrad: The sequence $$1 \rightarrow \mu \rightarrow Z' \times \mathcal{G} \rightarrow G' \rightarrow 1$$ is not exact in general. Namely, the kernel $\nu$ of the map $Z' \times \mathcal{G} \ri …

In addition to Jason's answer, I mention the following result, which I found out to be not known to experts (except Jason). Theorem. Let $X$ be a homogeneous space of a connected linear algebra …

Let $X$ be a smooth geometrically irreducible $k$-variety over a number field $k$. I do not assume that $X$ has a $k$-point. Is it true that $X$ has $k_v$-points for almost all places $v$ of $k$?

Let $\alpha$ be an algebraic integer on the unit circle in $\mathbb{C}$ such that all the conjugates of $\alpha$ lie on the unit circle. Does it follow that $\alpha$ is a root of unity?

Let $G$ be a (connected) reductive group over a perfect field $k$, and let $\xi\in H^1(k,G)$ be a cohomology class. By a theorem of Steinberg (Serre, Galois cohomology, Appendix 1 to Chapter III, Theo …

This is a partial answer inspired by comments of Jason Starr. I show that the answer is YES for $G=\mathrm{Sp}_{2n}$ (and also for the classical non-simply-connected groups $\mathrm{SO}_{2n+1}$ and $\ …

Find it here. (Edit: Link removed). I hope you can read Russian. Enjoy! Edit: Find here another (better?) scan, also in Russian.

Let $k$ be a number field. Let $M$ be a nonzero finite $\mathrm{Gal}({\bar k}/k)$-module, i.e., a nonzero discrete finite abelian group with a continuous action of $\mathrm{Gal}({\bar k}/k)$. Theor …

$\newcommand{\la}{\langle}\newcommand{\ra}{\rangle}$The following example is due to Vladimir Chernousov (private communication). Let $K={\Bbb Q}(x,y,x',y')$, where $x,y,x',y'$ are variables. Consider …

Let $k$ be a number field, and let $G$ be a split simply connected algebraic group over $k$. Let $\Omega_k$ denote the set of places of $k$. Let $T$ be a maximal torus of $G$ (defined over $k$). Cons …

See Subsection 6.4.1 in my paper with Zeev Rudnick Hardy-Littlewood varieties and semisimple groups, Invent. Math. 119 (1995), 37-66, page 62 (or this PDF file, page 23). The equation is: $$ -9x^2+2x …

Let $K$ be an arbitrary field, and let $K^s$ be a fixed separable closure of $K$. Let $F/K$ a be a finite Galois extension in $K^s$. Let $n>0$ be a natural number. Let $A$ be a central simple algebra …

Let $K$ be a number field, and let $S=S_f\cup S_{\mathbb R}\cup S_{\mathbb C}$ be a finite set of places of $K$, where $S_f$ denotes the set of finite places in $S$, $S_{\mathbb R}$ denotes the set of …

If the $G_K$-action on $M$ is trivial, then $$H^1(K,M)=\mathrm{Hom}(G_K,M),$$ and by Chebotarev's density theorem $$ F^1(K,M)=0.$$ For details see Lemma 1.1(i) of Sansuc, J.-J. Groupe de Brauer et ari …