# Tagged Questions

**3**

votes

**0**answers

73 views

### Given a positive-definite integral unimodular Gram matrix, how to find a basis of the associated lattice (over $\mathbf Q$)?

Let $G$ be a $n\times n-$symmetric matrix with integral coefficients and determinant $1$ (i.e. unimodular) such that the associated quadratic form is positive-definite.
I am interested in having an ...

**6**

votes

**0**answers

117 views

### Upper bound on the number of ismorphism classes of bilinear forms on $\mathbb{Z}^n$

$\DeclareMathOperator{\Hom}{Hom}$A symmetric, positive definite bilinear form on $\mathbb{Z}^n$ is any mapping $$b : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{Z}$$ satisfying
$b$ is bilinear,
...

**5**

votes

**1**answer

309 views

### Easiest way to distinguish $E_8 \oplus E_8$ from $E_{16}$

For $n$ an integer divisible by $8$, let me denote by $E_n$ the "usual" even non-degenerate positive definite integral symmetric bilinear form over $\mathbf Z^n$.
It is well known that in dimension ...

**1**

vote

**0**answers

57 views

### Is this related to a simple property of a lattice?

I am looking for a certain notion of sparseness of lattices.
I want to find a vector in $\mathbb{Z}^N$ that the minimal possible inner product with all the vectors of a given lattice. Or at least, I ...

**10**

votes

**3**answers

411 views

### 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 ...

**0**

votes

**1**answer

73 views

### A description of the isometry group $O(U\oplus E_8)$?

Are there any good description of the isometry group $O(U\oplus E_8)$? Here $U$ denotes the hyperbolic lattice and $E_8$ the root lattice of type $E_8$.

**2**

votes

**1**answer

260 views

### How to determine $O(L)$ is finite or not?

Let $L$ be an indefinite {\it non-unimodular} integral lattice. I am particularly interested in unimodular cases, such as $U(2)\oplus A_4, U\oplus D_4$. Are there any general method to determine ...

**2**

votes

**1**answer

283 views

### Automorphism groups of indefinite non-unimodular integer lattices

Does anyone know of any papers in which structural aspects of the orthogonal group of some indefinite non-unimodular integral lattice are calculated? The exact lattice isn't so important and they ...

**1**

vote

**0**answers

49 views

### Tensor product with $\mathbb{R}$ of an even unimodular lattice

Let $\Lambda$ be an unimodular even lattice of signature $(m,n)$.
By a classifying theorem by Milnor, $\Lambda$ must be of the form $U^k\oplus E_8(\pm 1)^l$, where $U$ is the hyperbolic plane.
Now ...

**6**

votes

**1**answer

402 views

### genus and spinor genus over a number field

Let $F$ be a number field with ring of integers $\mathfrak{o}$. Let $(V,Q)$ be a quadratic space of dimension $n$ over $F$, and let $L$ be a free lattice in $V$ (i.e. $L\cong\mathfrak{o}^n$). If the ...

**1**

vote

**0**answers

157 views

### Involution of $E_{8}$ lattice

Let $L$ be a lattice associate to the Dykin matrix of type $E_{8}$. I would like to understand involutions of $L$ and their invariant $L^{+}$ and coinvariant lattice $L^-$ (I think they are ...

**1**

vote

**1**answer

96 views

### Lorentz quotient and orientation

$$ U \; = \;
\left( \begin{array}{cc}
0 & 1 \\\
1 & 0
\end{array}
\right) ,
$$
Given a real oriented vector space $V$ with inner product, form Lorentzian $L = V \oplus ...

**5**

votes

**0**answers

159 views

### Effect of Covering Radius on Shortest Vector

For "even" integral lattices in dimension at least 4, does a covering radius strictly less than $\sqrt 2$ imply that there is a vector of norm 2, also called a root?
Note that this is simply false in ...

**2**

votes

**0**answers

212 views

### Hurwitz integers and $F_4$

The Hurwitz integers are
$$
\mathcal H=
\{a+bi+cj+dk:a,b,c,d\in\mathbb Z\;\text{ or } \;a,b,c,d\in \tfrac12+\mathbb Z\}.
$$
I want to know if there is a formula, for $m\in\mathbb Z$, for the number ...

**0**

votes

**0**answers

98 views

### Question regarding contiguous forms

I read about contiguous forms in Achill Scürmann's thesis on positive quadratic forms. I am wondering about one aspect of the Voronoi algorithm presented in there, that enumerates all arithmetically ...

**1**

vote

**2**answers

211 views

### About lattice $\pmod q$

For any matrix $A \in Z^{n\times m}$, Let $$\wedge_q(A)=\{ y\in Z^m:\exists s\in Z^n, s.t. y=A^ts (\mod q) \},$$ $$\wedge_q^\bot(A)=\{x\in Z^m: Ax=0 (\mod q)\}.$$ There is a result stating that ...

**5**

votes

**1**answer

419 views

### Finding Generators of O( Z^3,x^2 + xy + y^2 - z^2) and integer solutions

All pythagorean triples can be generated from $(3,4,5)^T$ and $(5,4,3)^T$ by the matrices:
\[ A = \left(\begin{array}{ccc} -1 & 2 & 2 \\\\ -2 & 1 & 2 \\\\-2 & 2 & 3 ...

**4**

votes

**1**answer

434 views

### Lorentzian characterization of genus

Suppose we take the "even" indefinite lattice from page 50 in Serre A Course in Arithmetic (1973)
$$ U \; = \;
\left( \begin{array}{cc}
0 & 1 \\\
1 & 0
\end{array}
\right),$$
...

**17**

votes

**3**answers

1k views

### A Priori proof that Covering Radius strictly less than $\sqrt 2$ implies class number one

It turns out that each of Pete L. Clark's "euclidean" quadratic forms, as long as it has coefficients in the rational integers $\mathbb Z$ and is positive, is in a genus containing only one ...

**6**

votes

**2**answers

407 views

### Is the square of the covering radius of an integral lattice/quadratic form always rational?

This is one of many observations from Pete L. Clark's questions on "Euclidean" quadratic forms. I sent Pete many positive integral forms that obeyed his condition. In turn, his condition turns out to ...