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.

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.

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.

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.