# Automorphic quotients for inner forms or $GSp(4)$

For a quaternion algebra $D$, introduce the quaternionic similitude unitary groups: $$\mathrm{GU}_D = \left\{ g \in \mathrm{GL}(D) \ : \ g^\star \left( \begin{array}{cc} & 1 \\ 1 & \end{array} \right) g = \mu(g) \left( \begin{array}{cc} & 1 \\ 1 & \end{array} \right), \mu(g) \in \mathbf{G}_m \right\}$$

The $\mathrm{GU}_D$ are the inner forms of $\mathrm{GSp(4)}$ when $D$ describes the quaternion algebras. For almost every places $v$ of $F$, more precisely the split ones, this group is $G_v \simeq \mathrm{GSp(4,F_v)}$.

When are those $\mathrm{GU}_D$ of compact automorphic quotient?

I believe assuming $D$ to be totally definite at infinity (i.e.ramified at archimedean places) is enough, but is it necessary?

• Actually, I don't think the form you write down is ever compact (mod center). You should probably take the form given by the identity matrix, or put a minus sign on one of the 1's. – Kimball Oct 13 '17 at 2:28
• @Kimball Yes I agree now, following Aurel's answer. I was thinking in varying the quaternion algebra, forgetting to let the quadratic form vary also ;) – Desiderius Severus Oct 16 '17 at 10:16

Let me give you the inner forms of $\mathrm{GSp}(4)$ with compact adelic quotient.

Every nonsplit inner form of $\mathrm{GSp}(4)$ is obtained in the following way: take $D$ a division quaternion algebra over $F$, let $V$ be a $D$-Hermitian space of $D$-dimension $2$, and construct $G = \mathrm{GU}(V)$.

By a theorem of Borel and Harish-Chandra, the adelic quotient of $G$ is compact if and only if $G$ modulo its center is anisotropic (in the algebraic group sense: it contains no nontrivial split torus).

This turns out to be equivalent to $V$ being anisotropic (in the quadratic/hermitian form sense: it has no nonzero isotropic vector). Over a number field, by the local-global principle for quadratic forms we can test anisotropy locally: over $p$-adic places, $V$ has $8$ variables as a quadratic form and is therefore isotropic, and over real places, $V$ is anisotropic if and only if it is positive definite or negative definite.

In summary, $G = \mathrm{GU}(V)$ has compact adelic quotient if and only if $F$ admits a real place at which $D$ is definite and $V$ is positive definite or negative definite.

For instance, the $D$-Hermitian form $x\bar{x}+y\bar{y}$ works over a totally definite quaternion algebra, but the one you wrote down is $x\bar{y}+y\bar{x}$.

• Note that I have interpreted your "automorphic quotient" as being the adelic double quotient of the adjoint group of $G$. This is probably the one you wanted. – Aurel Oct 10 '17 at 21:32
• This is pretty much what I wanted to understand, thank you. Is there any standard reference where I can find that description for inner forms of GSp(4)? – Desiderius Severus Oct 11 '17 at 5:59
• It is probably all in Platonov and Rapinchuk's "Algebraic groups and number theory". – Aurel Oct 11 '17 at 6:43

As you write it, this group always has a non-trivial parabolic subgroup (the Siegel subgroup, given by the intersection of $GU_D$ with upper-triangular matrices in $GL_2(D)$); and hence its symmetric space cannot be compact.

• However, is this group always a non-trivial one? I am thinking of $D$ totally definite at infinity, it gives rise to a compact group at archimedean places? And it suffices to have compacity at one archimedean place to have a compact automorphic quotient, or am I mistaken? I am nearly convinced by what you said, however I found this article of Taibi (hal.archives-ouvertes.fr/hal-00675682/document) and in Theorem B, p.6, he considers an inner form of the symplectic group which is compact at archimedean places... so I should be missing something! – Desiderius Severus Oct 9 '17 at 13:58
• Compact forms at the archimedean place exist, but they are not of the form that you describe in your question, even if $D$ is definite. The problem is that you are describing the unitary group of the isotropic $D$-Hermitian form. – Aurel Oct 9 '17 at 14:46
• @Aurel Oh, so that my question should be (and I am already glad to have clarified this point): are there any inner form of GSp(4) of compact quotient? – Desiderius Severus Oct 10 '17 at 20:33