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.
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asked Nov 26, 2009 at 9:10
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3$\begingroup$ You should probably ask this is some MAGMA forum... $\endgroup$– Mariano Suárez-ÁlvarezCommented Nov 26, 2009 at 12:16
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$\begingroup$ Or e-mail Harris Nover...he knows all this stuff. $\endgroup$– Ben WeissCommented Mar 20, 2010 at 2:35
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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."
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$\begingroup$ Yes by strong approximation the spinor genus and the isometry class are the same for indefinite forms( at least in dimension bigger than 2). The problem is that I don't know how to check whether the spinor genus of M and N is the same. If M is indefinite MAGMA does not accept LatticeWithGram(M). $\endgroup$ Commented Nov 27, 2009 at 4:42
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$\begingroup$ OK, if you really want to use MAGMA, you can use some trick like trying to replace M,N by p-adically close but definite forms M', N'; then use Magma's implemented functions to check if these are equivalent at p. (You do this at all primes p dividing the discriminant.) I'm not sure if Magma's implemented functions cover the "spinor" version of this, however. $\endgroup$– moonfaceCommented Nov 27, 2009 at 7:17