I've come across the claim "$ U_3(\mathbb{Q}) $ has class number $ 1 $."
I've also heard about $ U_n $, or really $ PU_n $, that "up to $ n=8 $ we can find class number $ 1 $ groups but above $ 8 $ we cannot." Apparently the impossibility for $ n > 8 $ follows from Prasad's volume formula http://www.numdam.org/item/PMIHES_1989__69__91_0.pdf All of this is related to "schemes of groups of class number $ 1 $."
How do you calculate the class number of a unitary group? What is the class number of a scheme of groups? Where does this bound $ n \leq 8 $ come from?
Context: These quotes are from papers and lectures by Shai Evra for example https://arxiv.org/pdf/1810.04710.pdf
Disclaimer: I'm very new to the concept of class number $ 1 $ and how it relates to groups so its very possible that I have incorrectly phrased a lot of these questions, but I did try to rely as much on direct quotes so things will make sense. I'm not even sure what to tag this post with because I know so little about this area.
I'm mostly looking for relevant references but I would be very interested if anyone has some thoughts or explanations about the claims above.
Let $G$ be a reductive group over $\mathbb{Q}$ such that $G(\mathbb{R})$ is compact, and let $K$ be an open compact subgroup of $G(\mathbb{A}_{\mathrm{f}})$. Then the double coset space $$G(\mathbb{Q}) \backslash \, G(\mathbb{A})\, / \left( K \cdot G(\mathbb{R})\right)$$ is finite, and its cardinality is often referred to as the class number of $K$. If $G$ comes with a natural $\mathbb{Z}$-scheme model $\mathcal{G}$, then the "class number of $\mathcal{G}$" is the class number of the open compact $K = \mathcal{G}(\widehat{\mathbb{Z}})$.
(See e.g. formula (3.8) of the arxiv preprint in your link, which states that "class number one" is equivalent to having $U_3(\mathbb{A}) = U_3(\mathbb{Q})U_3(\mathbb{R}) U_3(\widehat{\mathbb{Z}})$.)
As for how you compute these numbers: there is a quantity called the mass, which is related to the class number but is easier to work with. It's defined as the sum $$ \sum_{g \in G(\mathbb{Q}) \backslash \, G(\mathbb{A})\, / \left( K \cdot G(\mathbb{R})\right)} \frac{1}{\#\Gamma_g},\qquad \Gamma_g = G(\mathbb{Q}) \cap g K g^{-1}.$$ There are formulae for the masses of "nice" compact opens in terms of special values of $L$-functions; and if you compute the mass of the open compact $\mathcal{G}(\widehat{\mathbb{Z}})$ and it turns out to be equal to $1/\#\mathcal{G}(\mathbb{Z}) = 1/ \Gamma_{\mathrm{id}}$, then there can't be any more terms in the sum, so the class number must be 1. (This is pretty much what is going on from formula (3.8) to the end of the proof in the arxiv preprint.)