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Let $G$ be a simply connected semisimple group over a perfect field $k$ (at the moment I am interested in the case $k=\mathbb R$). Then $G$ is an inner form of a quasi-split $k$-group $G_{\rm qs}$: th …

Let $m\ge 2$, and let $G={\rm SO}^*(4m)$ denote the "quaternionic" real form of the special orthogonal group ${\rm SO}(4m,\mathbb C)$ of type ${\sf D}_{2m}$. Let $\tau\in{\rm Aut}_{\Bbb R}(G)$ be a re …

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 $\ …

I don't think so. I think a gerbe bound by $A$ over the spectrum of a field $k$ gives a cohomology class $\eta\in H^2(k,A)$, and the gerbe trivializes over an extension $k'/k$ if and only if this c …

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 …

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 …

$\require{AMScd} $In this answer, $k$ is a nonarchimedean local field. Lemma 1. Consider a short exact sequence of linear algebraic $k$-groups $$ 1\to C\overset i\longrightarrow G\overset j\longright …

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 …

Let $k$ be a number field and $M$ be a nonzero finite discrete $\mathrm{Gal}(\bar k/k)$-module. Is it true that $H^1(k,M)$ is infinite? This would complete the answer of Daniel Loughran. There is a c …

Let $k$ be a field of characteristic 0 with a fixed algebraic closure $\bar k$ and absolute Galois group $\Gamma={\rm Gal}(\bar k/k)$. Let $M$ be a finite $\Gamma$-module, that is, a finite abelian g …

We denote by $F^{\mathbb{R}}_{p,q}$ the quadratic form over the field ${\mathbb{R}}$ $$ F^{\mathbb{R}}_{p,q}(x)=x_1^2+\dots+x_p^2-(x_{p+1}^2+\dots+x_{p+q}^2) $$ on the vector space $V^{\mathbb{R}}:={ …

YES to Question 1. For an arbitrary homomorphism $\phi\colon G\to F$ of connected algebraic groups over $k$, not necessarily affine, where $k$ is a $p$-adic field or $k=\mathbb{R}$, the image $\phi(G( …