Automorphism Group – Orbits for Indefinite Lattices

complex-geometrylatticeslie-groupsnt.number-theoryquadratic-forms

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.

Best Answer

It is in Kneser's book Quadratische Formen. For each r, there are only finitely many classes of representations of r by the lattice.

Related Solutions

[Math] Lattices: why require bilinear form to be integral

One problem that seems to be implicit in your question is that the term "lattice" is used in many contexts, and has multiple definitions. Among people who work with integral bilinear forms or quadratic forms, the norm on a lattice is defined to take values in integers, but it is definitely not assumed to be positive definite (contrary to the definition you offered at the top).

The most classical arithmetic origins come from Brahmagupta's 7th century work on binary quadratic forms, leading to Gauss's composition law and questions related to Waring's problem about which natural numbers are represented by quadratic forms of a certain type. As others have mentioned, there are applications that arise quite naturally in studying the cohomology (in particular, intersection theory) of manifolds, and in number theory proper, where they arise in the study of modular forms via theta constants and trace forms. If you want to study orthogonal groups or Clifford algebras in a setting that includes both real coefficients and finite fields, it is necessary to consider quadratic forms defined over number rings, and in particular, lattices that take values in the integers. A more recent application is in lattice conformal field theory, where you need the bilinear form on a lattice to be integer-valued to yield a super vector space of states (and it must be even to yield an honest vector space).

In the case of complex elliptic curves, the integrality of the lattice (after suitable rescaling) is equivalent to the elliptic curve having complex multiplication. There are plenty of interesting things to say about how these curves relate to class field theory, but it doesn't have much to do with the curves being defined over $\mathbb{Q}$. Most curves over $\mathbb{Q}$ do not have CM, and most CM curves are not defined over $\mathbb{Q}$. See chapter 2 in Silverman's Advanced Topics.

Number Theory – Integral Orthogonal Group for Indefinite Ternary Quadratic Form

Edit : this is a new answer, after more computations.

Let $H$ be the subgroup of your orthogonal group that preserve globally each connected component of the (two-sheeted) space $q(x,y,z)=-1$.

Up to this action, there is a single isometry class of isotropic vectors.

One representant is $(-1\ -3\ 8)$ and its stabilizer is the infinite dihedral group generated by

B1 = [ -39 -152  380]   C1 = [-2889 -8398 22610]
     [ -40 -151  380]        [-2210 -6421 17290]
     [ -20  -76  191]        [-1190 -3458  9311]

The group $H$ can be described as follows : let us write

A1 = [-1  0  0]             B1 = [ -39 -152  380]             B6=[ -609 -1216  3800]
     [ 0  1  0]                  [ -40 -151  380]                [ -320  -641  2000
     [ 0  0  1]                  [ -20  -76  191]                [ -200  -400  1249]

A2 = [ 1  0  0]             B2 = [-23751 -50350 152950]       T=[-609  -76 2660]
     [ 0 -1  0]                  [-13250 -28091 85330]          [-380  -49 1660]
     [ 0  0  1]                  [ -8050 -17066 51841]          [-220  -28  961]

A3 = [ 1  0  0]             B3 = [-1608009 -3459026 10438030]
     [ 0 -9 20]                  [ -910270 -1958101  5908810]
     [ 0 -4  9]                  [ -549370 -1181762  3566111]

A4 = [-170    0  741]           B4 = [-194561 -415872 1258560]
     [   0    1    0]                [-109440 -233929  707940]
     [ -39    0  170]                [ -66240 -141588  428489]

A5 = [ -930 -2128  6251]        B5 = [-39 -38 190]
     [ -560 -1279  3760]             [-10 -11  50]
     [ -329  -752  2210]             [-10 -10  49]

All the matrices $Ai$ and $Bi$ have order 2, while $T$ has infinite order.

Let $K$ be the free product of all the subgroups generated by these element (thus it is isomorphic to $\mathbf Z/2^{\star 11}\star \mathbf Z$.

Let $R$ be the subgroup of K generated by $[A1,A2]$ , $[A1,A3]$, $[A3,B1]$, $[A2,A4]$, $[A5,B3]$, and $T.A4.T^{-1}.A5$.

Then $R$ is the kernel of the obvious representation of $K$ in $H$, which is an epimorphism.

The group $\mathrm H_1(H,\mathbf Q)$ has dimension 1, thus $H$ is not a reflective group. I hope to be able to draw a Coxeter Diagram for its reflection subgroup soon. If this happens, I will edit this answer once more.

Best Answer

Related Solutions

[Math] Lattices: why require bilinear form to be integral

Number Theory – Integral Orthogonal Group for Indefinite Ternary Quadratic Form

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