In some cases the relations between a global group $G$ (over the adeles $\mathbb{A}$ of a field $F$) and its local components $G_v$ (where $v$ are the places of $F$) are well known. Obviously a group determines its local components, so the question aims at, given a family of local groups , understanding whether or not it can arises as the local components of global group. Some examples:

- the local components of a quaternion algebra are $GL_2(F_v)$ almost everywhere, and an even number of local division quaternion algebras $B_v$
- the local components of a unitary group are $GL_2(F_v)$ at the split places, and any collection of local unitary groups at the other places, maybe safe one (Hasse invariants condition, empty if the number of variables is odd)

Now here is the question:

what natural global group can have $GSp(4)$ as (one, many or every) local component(s)?