Let $F$ be a (totally real) number field, and $E$ a (totally imaginary) quadratic extension of $F$. We consider $U$ a unitary group (with respect to a given hermitian form over $E$). The question is:



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> When is $U(F) \backslash U(\mathbf{A}\_F)$ compact?
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I found some interesting reflexions about that in Arthur's notes on the trace formula, mainly that generally $U(F) \backslash U(\mathbf{A}\_F)$ cannot be compact for it contains some $\mathbf{R}^k$ embedded diagonally, corresponding to the archimedean places. It is what happens for general linear groups, even for the case of $\mathrm{GL}\_1$.


So I believe I have to assume that the $U\_v$ are compact (that is, the usual $U(n)$) at archimedean places, and then answer the question for $U(F) \backslash U(\mathbf{A}\_F)^1$. Is it enough to suppose so? Arthur cites a general characterization due to Borel-Harish-Chandra as well as Mostow-Tamagawa:



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> $U(F) \backslash U(\mathbf{A}\_F)^1$ is compact if and only if $U$ has no proper parabolic subgroup defined over $\mathbf{Q}$
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and that seems to be equivalent to $U(F)$ having no unipotent in his radical (is that what we call anisotropic?).


My current interest is in the unitary group in **two variables**, and I can suppose it to be definite, that is **compact at archimedean places**. Is it at least true in this case, and is there a simpler and more direct proof of this fact in this case, without invoking such a general result?



Compactness of the automorphic quotient
There are actually several different conditions all of which are equivalent, and one can make various formulations of these conditions depending on whether one sticks to the reductive case and/or global fields of characteristic 0 vs. general global fields. So it may be confusing to sort out exactly what is equivalent to what in which circumstances, and which equivalences are really theorems in number theory rather than instances of general equivalences over arbitrary fields. One thing which should be stressed at the outset, however, is that compactness at archimedean places is very very far from necessary for any such compactness criterion (though it is sufficient in the connected reductive case, as is compactness of the group of points at *any single place*, due to the various equivalences discussed at the end below).


Below I give a summary of the general situation with references for proofs.


For any connected reductive group $G$ over any field $k$ whatsoever, the absence of proper parabolic $k$-subgroup is equivalent to the condition 


$(\*)$: all split $k$-tori $T \subset G$ are central (equivalently, the connected semisimple derived group $\mathscr{D}(G)$ is $k$-anisotropic)


(see 20.6(ii) in Borel's textbook on algebraic groups for a proof). 


Example: If $(V,h)$ is a non-degenerate finite-dimensional hermitian space over $k$ attached to a separable quadratic extension $k'/k$ then ${\rm{U}}(h)$ satisfies $(\*)$ if and only if ${\rm{SU}}(h)$ is $k$-anisotropic, and that in turn is equivalent to the condition that the associated non-degenerate quadratic form $q\_h:v \mapsto h(v,v)$ on $V$ is non-vanishing on $V-\{0\}$. This is proved similarly to the analogous fact that if $(W,q)$ is a non-degenerate finite-dimensional quadratic space over $k$ then ${\rm{SO}}(q)$ is $k$-anisotropic if and only if $q$ is non-vanishing on $V-\{0\}$.


Going beyond the reductive case (e.g., smooth connected solvable groups), for an *arbitrary* smooth connected affine $k$-group $G$ one can also consider the condition


$(\*)$': all split $k$-tori $T \subset G$ satisfy $T\_{\overline{k}} \subset \mathscr{R}(G\_{\overline{k}})$.


In the reductive case, $(\*)$' is obviously equivalent to $(\*)$ because $\mathscr{R}(G\_{\overline{k}})$ is the maximal central $\overline{k}$-torus of $G\_{\overline{k}}$ which we know always descends to the maximal central $k$-torus of $G$ (obvious when $k$ is perfect by Galois descent, and part of the structure theory of reductive groups for general $k$).
Beyond the reductive case these two conditions are obviously not equivalent (consider solvable $G$). 


Although the $\overline{k}$-subgroup $\mathscr{R}(G\_{\overline{k}}) \subset G\_{\overline{k}}$ does descend to a $k$-subgroup of $G$ when $k$ is perfect (by Galois descent), so for such $k$ we can remove the intervention of $\overline{k}$ in $(\*)$', what about in general? Beyond the reductive case whenever $k$ is not perfect there are many examples where $\mathscr{R}(G\_{\overline{k}})$ does not descend to a $k$-subgroup of $G$. The theory of pseudo-reductive groups provides zillions of such examples, of which the most concrete are smooth connected affine Weil restrictions $G:= {\rm{R}}\_{k'/k}(G')$ for non-separable finite extensions $k'/k$ and non-trivial connected semisimple $k'$-groups $G'$; such $G$ are never reductive. 


So if we define the so-called *$k$-radical* $\mathscr{R}\_k(G)$ to be the maximal solvable smooth connected normal $k$-subgroup of $G$ then typically $\mathscr{R}\_k(G)\_{\overline{k}}$ is a proper $\overline{k}$-subgroup of $\mathscr{R}(G\_{\overline{k}})$ (e.g., it is trivial for the non-reductive Weil-restriction examples just mentioned). But despite that, one might wonder nonetheless if for the purpose of containing split tori it may not matter: perhaps $(\*)$' is equivalent to the condition


$(\*)$'': all split $k$-tori $T \subset G$ satisfy $T \subset \mathscr{R}\_k(G)$.


Indeed, it turns out that the equivalence holds in general. The proof requires a digression into the structure of pseudo-reductive groups, which I suspect you don't care about, so I'll refrain from saying more on that here. 


Moving on from tori to unipotent groups, recall that a smooth connected unipotent $k$-group $U$ is called *$k$-split* if it admits a composition series by smooth connected $k$-subgroups such that the successive quotient is $\mathbf{G}\_a$; e.g., for perfect $k$ this always holds (see 15.5(ii) in Borel's textbook on linear algebraic groups). There is yet another condition one could consider, on the unipotent side:


(\*\*)': every $k$-split smooth connected unipotent $k$-subgroup $U \subset G$ satisfies $U\_{\overline{k}} \subset \mathscr{R}(G\_{\overline{k}})$.


Note that in characteristic 0 the group $U(k)$ is Zariski-dense in $U$, and if $u \in G(k)$ is a nontrivial unipotent element then the Zariski closure of the $k$-group it generates is smooth connected unipotent and of course $k$-split. Hence, *in characteristic $0$* the condition $(\*\*)$' is equivalent to the condition (mentioned in the question posed) that all unipotent elements of $G(k)$ are contained in the geometric radical. It is asked in the question posed if this is what we call "$k$-anisotropic", which is really defined as the absence of nontrivial split $k$-tori. For connected semisimple groups in characteristic 0 it is true that the absence of nontrivial unipotent $k$-points is equivalent to $k$-anisotropicity, but in characteristic $p>0$ this is a rather more subtle issue since unipotent $k$-points have finite order and so it isn't evident if they must lie inside smooth connected unipotent $k$-subgroups, let alone ones which are $k$-split; let's not dwell on that any further here.


Inspired by the success with $(\*)$'', one might wonder if there is a unipotent analogue: if we define the *$k$-unipotent radical* $\mathscr{R}\_{u,k}(G)$ to be the maximal unipotent smooth connected normal $k$-subgroup of $G$, one can ask if $(\*\*)$' is equivalent to the condition


$(\*\*)$'': every $k$-split smooth connected unipotent $k$-subgroup $U \subset G$ satisfies $U \subset \mathscr{R}\_{u,k}(G)$.


Obviously the equivalence holds when $k$ is perfect or $G$ is reductive. It does hold in general, but again that involves some (quite elementary) considerations in the direction of pseudo-reductive groups, so we'll not say more about it here. 


Borel and Tits announced that $(\*)$' and $(\*\*)$' are actually equivalent (!) in complete generality. In the connected reductive case this is more familiar via the link to proper parabolic $k$-subgroups, but for imperfect $k$ it is a rather delicate matter with $(\*\*)$' (i.e., showing that the maximal $k$-split unipotent smooth connected $k$-subgroups are precisely the $k$-unipotent radicals of the minimal parabolic $k$-subgroups). It seems that Borel and Tits never published a proof of the general equivalence, but a proof is given as Proposition A.5.1 in the paper *Finiteness theorems for algebraic groups over function fields* in Compositio 148 (2012) (using in a crucial way the structure theory of pseudo-reductive groups). 


Finally, having stated this blizzard of various conditions over arbitrary fields (no number theory yet!), we reach the main point. Let $G$ be an arbitrary smooth connected affine group over a global field $k$. As usual, define $G(\mathbf{A}\_k)^1$ to be the subgroup of points $g \in G(\mathbf{A}\_k)$ such that $|\!|\chi(g)|\!|\_k = 1$ for all $k$-homomorphisms $\chi:G \rightarrow \mathbf{G}\_m$. Godement conjectured for global fields $k$ (or at least he did so in characteristic 0) that $G(k)\backslash G(\mathbf{A}\_k)^1$ is compact if and only if $G$ satisfies both $(\*)$' and $(\*\*)$' that we have noted above are equivalent to each other over general fields. 


As is discussed with references early in section A.5 of the Compositio paper mentioned above, the conjecture of Godement was proved over number fields independently in joint work of Borel with Harish-Chandra, and of Mostow with Tamagawa, and over function fields the reductive case was settled by Harder. In case you might wonder about the case of function fields for general $G$ (i.e., not necessarily reductive), the necessity of $(\*)$' (or equivalently $(\*\*)$') is due to Oesterle (see 1.4 in Chapter IV of his paper *Nombres de Tamagawa et groupes unipotents en caract\'eristique $p$*, Inventiones Math. 78 (1984)) but the sufficiency in general again requires the structure theory of pseudo-reductive groups; see Theorem A.5.5 in the Compositio paper mentioned above.


