The number of quadratic forms attaining Hermite's constant

Question

$\require{AMScd}$ I'm considering minimum values (at non-zero integer points) of real, positive-definite, quadratic forms of determinant $1$. These are functions $f:\mathbb{R}^n\to \mathbb{R}$ which can be represented as $x\mapsto x^\intercal Ax$ for some symmetric, positive-definite matrix $A$ with determinant $1$. Call this function space $\mathcal{Q}_n$.

Since I'm interested in extreme values (at integer points), I'm going to identify two such functions $f \sim f'$ if there is some invertible integer matrix $\Gamma$ such that $f'(\cdot)=f(\Gamma \cdot)$. This gives me a set $\mathcal{Q}_n/\sim$.

Now (using Hermite or Minkowski) one can easily demonstrate uniform (over $f$) bounds such as $$\min\limits_{v\in \mathbb{Z}^n\smallsetminus 0}f(v) \leq \left(\frac{4}{3}\right)^{(n-1)/2}, $$ so that $$S_n:=\sup\limits_{f} \left\lbrace\min\limits_{v\in \mathbb{Z}^n\smallsetminus0}f(v)\right\rbrace $$ is finite. Moreover, one can show (by a compactness argument due to Mahler) that in each dimension $n$, this bound is achieved by some particular quadratic form.

Question: In each dimension $n$, is there a finite bound on the equivalence classes $[f] \in \mathcal{Q}_n/\sim$ that can attain this maximum? That is, can we have an infinite number of inequivalent $f$ for which $$\inf\limits_{v\in \mathbb{Z}^n\smallsetminus0}f(v) = S_n? $$

For example, in dimension $2$ (resp. $3$) Hermite (resp. Gauss) showed that this max is only attained by (an appropriate multiple of) $[x^2+xy+y^2]$ (resp. $[x^2+y^2+z^2+xy+xz+yz]$).

Thanks for reading! Sorry if this question is not appropriate.

Answer 1

There are only finitely many inequivalent forms which may be local maxima for the Hermite invariant. Voronoi showed (1908) that the lattices attaining a local maxima are extreme, i.e. perfect and eutactic. Voronoi also showed there are only finitely many perfect lattices in each dimension, and these must all be integral by an older result of Korkine and Zolotarev (1877). There does not appear to be any known explicit bounds on the number of inequivalent extreme lattices holding for all dimensions. The forms which attain Hermite's constant are unique for $d\leq 8$ and these are all root lattices (see Conway and Sloane, Chapter 6 for more details). For more details on this, Chapter 3 of Martinet's book Perfect Lattices in Euclidean Spaces may be useful.

All of the perfect lattices/forms have been classified up to dimension eight and the number of these lattices in dimensions $d=1,2,3,4,5,6,7,8$ are $1, 1, 1, 2, 3, 7, 33, 10916$ respectively; the last number is known thanks to the work of Sikirić, Schürmann and Vallentin. The approach here is essentially algorthmic, (using Voronoi's algorithm) and in principle, given enough computational resources it is possible to enumerate all such forms in a given dimension. There is an ongoing attempt to classify all $9$-dimensional perfect lattices where the number of such found is already in the millions.