I have been studying the representation of integers by sums of squares. Consider the equation $$x_1^2+\cdots+x_n^2=k,$$ and let $r_n(k)$ denote the number of integer solutions. One way to compute $r_n(k)$ is to use the circle method of Hardy, Littlewood, and Ramanujan, which yields the asymptotic $$r_n(k)=\frac{\pi^{\frac n2}k^{\frac n2-1}}{\Gamma\left(\frac n2\right)}\sum_{q=0}^\infty A_q+O\left(k^{\frac n4}\right),$$ where $\Gamma$ is the usual Gamma function, $$A_q=q^{-n}\sum_{\substack{a\bmod q \\ (a, q)=1}}\left(\sum_{x=0}^{q-1} e\left(-\frac{ax^2}{q}\right)\right)e\left(-\frac{ak}{q}\right),$$ with $e(z)=e^{2\pi iz}$. This approach only works for $n\geq 3$, and the error term is known to be identically zero for $3\leq n\leq 8$. But for $n>8$, there are times when the error term is nonzero, and by looking at the Smith-Minkowski-Siegel mass formula, it seems that the obstruction is that the genus of the quadratic form $x_1^2+\cdots+x_n^2$ has more than one equivalence class for certain values of $n>8$.
Is this observation true? I would think that the genus of this particular form should contain precisely one class, namely $SO(n, \mathbb Z)$, yet the above observations suggest otherwise. It's not at all clear to me why this would even depend on $n$, especially since the condition that $n>8$ seems sort of random. When is the genus of $x_1^2+\cdots x_n^2$ bigger than just one class?
Watson's project of finding all genera of class number one, positive forms, was automated by Kirschmer and Lorch,. The maximum dimension is 10.
Watson proved in his 1962 Transformations paper that any genus of positive quadratic forms of dimension 11 or larger has class number larger than one.