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Let $F$ be a nonarchimedean local field of characteristic zero. Let $G=\operatorname{Res}_{E/F}\operatorname{GL}_n$ or $\operatorname{Res}_{E/F}\operatorname{U}_n$, where $\operatorname{U}_n$ is any unitary group of rank $n$. How many 2-fold linear algebraic covers of $G$ are there? Namely, how many linear algebraic groups $\widetilde{G}$ are there such that we have the following short exact sequence of group schemes? $$ 1\longrightarrow\mu_2\longrightarrow\widetilde{G}\longrightarrow G\longrightarrow 1 $$

Among those, which $\widetilde{G}$ has the property that $H^1(F, \widetilde{G})\neq 1$?

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  • $\begingroup$ What kind of extension is $E$? $\endgroup$
    – მამუკა ჯიბლაძე
    Commented Mar 15, 2021 at 7:09
  • $\begingroup$ Let us call ${\rm Ext}(G,\mu_2)$ the set of isomorphism classes of these extensions. I think that the homomorphism $G\to G^{\rm tor}$ induces an isomorphism $${\rm Ext}(G^{\rm tor},\mu_2)\overset{\sim}{\longrightarrow} {\rm Ext}(G,\mu_2),$$ where $G^{\rm tor}=G/(G,G)$ (the quotient by the commutator subgroup), because $(G,G)$ is simply connected. $\endgroup$
    – Mikhail Borovoi
    Commented Mar 20, 2021 at 19:44
  • $\begingroup$ Thus we come to ${\rm Ext}({\rm Res \,GL}_1,\mu_2)$ and ${\rm Ext}({\rm Res\,U}_1,\mu_2)$. I think that Res does not change Ext, and so we come to ${\rm Ext}({\rm GL}_1,\mu_2)$ and ${\rm Ext}({\rm U}_1,\mu_2)$. Each of them has cardinality 2, the nontrivial extensions being the standard double covers $$\rm GL_1\overset 2 \longrightarrow GL_1\quad\text{and}\quad \rm U_1\overset 2 \longrightarrow U_1\,.$$ Among those $H^1(F,\rm U_1)\ne 1$. $\endgroup$
    – Mikhail Borovoi
    Commented Mar 20, 2021 at 20:13

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