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Let $K$ be a field of characteristic zero. Let $G_K:=Gal(\bar{K}/K)$

The nontrivial elements of the set $H^1(G_K,PGL_2)$ correspond to $\bar{K}/K$-forms of $\mathbb{P}^1$; i.e. curves that are isomorphic to $\mathbb{P}^1$ over $\bar{K}$, but not over $K$.

Now let $D_\infty$ be the group $\mathbb{G}_m\rtimes\mu_2$ embedded in $PGL_2$ as $\{az:\,a\in \bar{K}^\times\}\cup\{b/z:\,b\in \bar{K}^\times\}$.

My question: Can one give a similar interpretation to the elements of $H^1(G_K,D_\infty)$?

That is: Find a scheme $X$ (I guess it will have $Aut(X)=D_\infty$) together with a nice map (perhaps it is only an injection)

$ \{$$\bar{K}/K$-forms of $X\}\to H^1(G_K,D_\infty)$.

Motivating reference: Silverman, Joseph H.. "The field of definition for dynamical systems on $\mathbb {P}^1$." Compositio Mathematica 98.3 (1995): 269-304.

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    $\begingroup$ What is $z$ in your definition? $\endgroup$ Feb 20, 2014 at 21:13
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    $\begingroup$ @DanielLoughran he means the Möbius transformations $z \mapsto az$ and $z \mapsto b/z$, respectively, with $z$ a coordinate on $\mathbb P^1$. $\endgroup$ Feb 21, 2014 at 14:08

1 Answer 1

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You can take $X={\mathbb G}_m$ (considered as an affine curve). Then $Aut(X)=D_{\infty}$ and you will have a desired bijection. In this case it is not difficult to describe the set of all $\bar K/K-$forms of $X$: they are curves $x^2-dy^2=m$. For example for $K={\mathbb R}$ we have 3 such curves up to ${\mathbb R}-$isomorphism: $x^2-y^2=1$ and $x^2+y^2=\pm 1$.

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