orbits of automorphism group for indefinite lattices I have a question about indefinite lattices.
QUESTION: Let $\Lambda\times\Lambda\rightarrow {\Bbb Z}$ be a lattice,
that is, ${\Bbb Z}^n$ with a non-degenerate integer quadratic form,
not necessarily unimodular, and $G:=O(\Lambda)$ the group of 
(integer) isometries. Denote the set of all vectors $v\in\Lambda$ such that $v^2=r$ by $S_r$. I think that it is true (under some 
additional assumptions on rank) that $G$ acts on $S_r$ with finitely 
many orbits, but I don't know a good reference. 
In this paper we have an argument proving this
for $r=0$: http://arxiv.org/abs/1208.4626
(Theorem 3.6), when the rank of a lattice 
is $\geq 7$. 
For unimodular lattices I think there is just
one orbit ("Eichler's theorem"), probably for rank $\geq 5$.
I would be very grateful for any reference
to this result in bigger generality, with
arbitrary $r$ and without unnecessary rank
restrictions.
The question comes from complex geometry: if $M$ is a hyperkahler manifold, there is a canonical non-unimodular integer quadratic form in $H^2(M)$, and its automorphisms are identified (up to finite index) with the mapping class group of $M$. Various geometric questions about $M$ are translated into lattice-theoretic questions about this lattice.
 A: It is in Kneser's book Quadratische Formen.  For each r, there are only finitely many classes of representations of r by the lattice.  
A: I think you are describing Siegel's construction for counting the representations of a number by a genus of quadratic forms. For positive forms, you count the number of representations by each class in the genus, but divide each one by the number of integer automorphs (isometries) of the particular form. In the end, you divide by the mass of the genus, the sum of the reciprocals of the automorph counts. For indefinite forms, all of that goes sideways, instead you count the orbits that you are describing, and this is finite, and not dependent on dimension. I will see if I can find a good description...
Found it, you want Schulze-Pillot_2004 at TERNARY, especially Siegel's Main Theorem on pages 305-306. Apparently this is also in Kitaoka's book.  
A: There's a simple criterion for equivalence of two vectors in a lattice if the lattice contains two copies of the hyperbolic plane: you look at the length of the vector and its image in the discriminant group of the lattice. Gritsenko, Hulek and Sankaran  mention it here http://arxiv.org/abs/0810.1614 but the result goes back to Eichler (they refer to it as the Eichler criterion). If you're working with the Beauville form, then all known Hyperkahlers satisfy the hyperbolic condition.
