I would like to understand precisely the structure of local unitary groups. A property that seems to appear in many places and which I do not understand is that a unitary group is ramified at half the places.
So let $F$ be a global non-archimedean number field, $E$ a quadratic extension of $F$, and $U$ a unitary group (i.e. the group of automorphism preserving a n hermitian form on $E$), say of rank $n$ (i.e. the hermitian form is on a squared $n$-dimensional vector space).
What can be said for the local groups, that is for the $U_v$ where $v$ goes through the places of $F$? More precisely:
- is there only a finite number of possibility for $U_v$?
- why "half" of the $U_v$ are "unramified", in the sense that $U_v \cong \mathrm{GL}_n(F)$?
Thanks in advance !