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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)?

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In addition to the relatively boring extension/restriction of scalars for $GSp(2n,\widetilde{k})$ for an extension field $\widetilde{k}$ of the ground field $k$... :

The Galois twist groups often denoted by $Sp^*(p,q)$ (or $GSp^*(p,q)$...) over ground field $\mathbb R$, defined via non-degenerate quaternion-symmetric forms with quaternion algebras over the ground field (or over extensions...) almost everywhere locally become $Sp(p+q,k_v)$, because the quaternion algebra splits almost everywhere locally. (Throwing in the similitudes is minor...)

That is, with quaternion (division) algebra $B$ over ground field $k$ (a global field), with non-degenerate $B$-valued, symmetric form on a $B$-vectorspace $V$ of $B$-dimension $n$, the isometry group $G$ has $k_v$-points isomorphic to $Sp(2n,k_v)$ locally almost everywhere.

EDIT: it may be worth adding that this sort of Galois twisting allows easy creation of compact quotients analogous to Shimura curves for $SL(2)=Sp(2)$, by choosing a symmetric form whose signature at one real place is $(0,q)$ or $(p,0)$, so that (by a standard, if not widely understood, reduction theory result) the arithmetic quotient is compact. But/and the representation theory is just that of (locally, split) $Sp(n)$ almost everywhere.

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    $\begingroup$ ... and see A. Weil's "Involutions and the classical groups" (or similar title) from about 1964 for proof that there are no other classical groups than what we can imagine by devices similar to my answer... I heard an exposition of this from G. Shimura on some day in the mid 1970s. $\endgroup$ Aug 10, 2017 at 0:59
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    $\begingroup$ This answer gives a more general result; if one is to confine oneself to $GSp_4=GSpin(5)$, matters are a bit easier to describe: take any non-degenrate quadratic form $f$ in 5 variables over $\mathbb Q$; then for almost all primes $p$, the group becomes the (quasi-split=split) group $GSpin(5)=GSp_4$. Over any (number) field, this (namely $GSpin (f)$) is the only way to get (inner) forms of $GSpin (5)$ . $\endgroup$ Aug 10, 2017 at 4:41
  • $\begingroup$ @Venkataramana Thanks for the answer, however when you say "the group", you are talking about the group of isometry defined for the quadratic form you consider? $\endgroup$ Aug 28, 2017 at 10:04
  • $\begingroup$ @paulgarrett You answer seems to be exactly the general answer I am seeking. However why is it clear that the isometry group is the symplectic group almost everywhere? Is it the case exactly at the split places? Do you know what happen in the ramified places? I am really not at ease with this kind of computations... $\endgroup$ Aug 28, 2017 at 10:23
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    $\begingroup$ @DesideriusSeverus, a quaternion division algebra splits almost everywhere locally, so the local group is the split $Sp(4)$ almost everywhere locally. At the (finitely-many) ramified places: at real ones, one has a "signature", while at finite ones there is a maximal possible size of isotropic subspace (as with $p$-adic quadratic forms). $\endgroup$ Aug 28, 2017 at 13:02

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