Consider a global field $F$ and the group $\Gamma =GL(2,F)$. An element $\gamma \in \Gamma$ is called elliptic, if its eigenvalues do not lie in $F$. Now consider a completion $F\_v$ of $F$ and $G\_v = GL(2,F\_v)$. Now the eigenvalues of $\gamma$ may or may not lie in $F\_v$. What is the centralizer
$$ C\_{G\_v}(\gamma) = \{ g \in G\_v : g\gamma = \gamma g \}?$$ 



Centralizer of elliptic elements in $GL(2)$
The centralizer of $\gamma$ is a torus of the form $E^\times$, where $E/F$ is a quadratic field extension. Assume moreover that $\gamma$ is regular (so that $E/F$ is separable). The centralizer of $\gamma$ in $G\_v$ is $(E\otimes\_F F\_v )^\times$. The algebra $E\otimes F\_v$ is either a field (in that case $\gamma$ is elliptic in $G\_v)$) or a sum of two fields isomorphic to $F\_v$, according to whether the prime $v$ splits in $E$ or not. In the latter case $\gamma$ is a regular element lying in a split torus of $G\_v$. 


