'Compute' Integral equivalence of matrices

Question

Hi.

For a matrix $D \in \mathbb{Z}^{n \times n}$ and a symmetric, positive definite integral even matrix $S \in \mathbb{Z}^{n \times n}$ put $S[D] := D^TSD$ where the $\cdot^T$ means 'transposed'. Further, for a symmetric matrix $X \in \mathbb{Z}^{n \times n}$ put $$R(S, X) := \{ D \in \mathbb{Z}^{n \times n} : S[D] = X \}$$

My question is the following: Can one give a more concrete description of the elements in it or can one at least compute this set in an efficient way (for relatively small $n$, say $\leq 20$ or so)? If this is not possible/unknown, does it maybe work with

$$R(S, X) / GL_n(\mathbb{Z})$$

or something like this? From computations in very small dimensions i got the impression that this set is actually relatively small.

One can show (see for example Koecher/Krieg, a german book on modular forms, p. 262) that $R(S, X)$ is finite and that

$$R(S, X) \subset \{ D \in \mathbb{Z}^{n \times n} : \sum_{i,j} |D_{i,j}|^2 \leq \alpha(S,X) + 1 \}$$

where $\alpha(S,X) = Trace(X)/\lambda$ and $\lambda$ is the minimal eigenvalue of $S$, i.e. $\alpha(S,X)$ can be easily computed. The problem is, of course, that the right hand set is pretty big even for dimensions $\leq 20$.

Cheers,

Fabian Werner

Answer 1

The Siegel mass formula gives an expression for a quantity closely related to the cardinality of $R(S,X)$. Perhaps you can use it to get a-priori bounds, so that you will know when to stop the calculation.

Unfortunately, computing everything in the Siegel mass formula (and especially getting the computation right) is a lot of work, so I am not really sure it is the way to go. You can look at the wikipedia article to get started, (even though most of it deals with the case where you represent an integer by a quadratic form, and not, as you want, the case where you represent one quadratic form by another quadratic form).