Question

Let's say we have two $n$-dimensional lattices $(V,b)$ and $(W,b_1)$ equipped with integral bilinear forms $b$ and $b_1$ respectively. Is there an implemented function in MAGMA that decides whether $(V,b)$ and $(W,b_1)$ are isometric? Equivalently given two symmetric $n \times n$ integer matrices $M$ and $N$, is there any function that decides if $T^{t}MT=N$ for some $T \in GL_{n}(Z)$. For positive definite $M$ and $N$ one can do it by defining LM:=LatticeWithGram(M) and LN:=LatticeWithGram(N) and then asking IsIsometric(LM,LN). Since the input of LatticeWithGram must be positive definite, the above does not work for indefinite matrices.

Answer 1

not an answer to the question, but: Checking isometry is much easier for indefinite forms; it's purely local, by strong approximation for the spin group. If interested search for "spinor genus."