# Questions tagged [lattices]

Lattices in the sense of discrete subgroups of Euclidean spaces, as used in number theory, discrete geometry, Lie groups, etc. (Not to be confused with lattice theory or lattices as used in physics! For lattices (ordered sets), use the tag: [lattice-theory])

646
questions

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votes

1
answer

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### Outer automorphisms of finitely generated linear groups

Is there an example of a finitely generated subgroup $\Gamma \subset \mathrm{GL}_n(\mathbb{C})$ such that the group of outer automorphisms $\mathrm{Out}(\Gamma)$ contains finite subgroups of unbounded ...

2
votes

1
answer

372
views

### Kissing number lower bound vs. upper bound - precise meanings?

According to en.wikipedia.org, https://en.wikipedia.org/wiki/Kissing_number#Some_known_bounds
It says the kissing numbers $K$ have lower bound $K_L$ and upper bound $K_S$:
$$
K_L < K < K_U.
$$
I ...

0
votes

1
answer

113
views

### Leech lattice shortest vector vs other 23 cases and E8 cases

In this paper by Viazovska, she said that:
"The E8-lattice sphere packing 𝒫E8 is the union of open Euclidean balls with centers at
the lattice points and radius $1/\sqrt{2}$." So I think ...

0
votes

0
answers

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views

### Reference on high dimensional polytopes with E8 symmetry

I'm starting to work with high dimensional polytopes. I am interested in uniform polytopes of 16-dimension and of 8-dimension (especially Elte and Gosset polytopes that have E8 symmetry).
...

7
votes

1
answer

284
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### Relation between different $E_8$ matrices

There are several rank-8 square matrices known to be related to $E_8$:
Cartan $E_8$ matrix https://en.wikipedia.org/wiki/E8_(mathematics)#Cartan_matrix
$$M_1=\left [\begin{array}{rr}
2 & -1 &...

4
votes

1
answer

388
views

### A note on orders in quaternion algebras

Definition. An algebra $B$ over a field $F$ is a quaternion algebra if there exists $i,j\in B$ such that $1,i,j,ij$ is a basis for $B$ as a vector space over $F$, where $i^2=a,j^2=b;a,b\in F^\times$.
...

2
votes

0
answers

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views

### decidability special case of column generation problem

I have the following problem:
Input: sub-spaces $V_1, \dots, V_d$ of $\mathbb{Z}^{d}$
Question: are there $v_i \in V_i$ such that the matrix $(v_1, \dots, v_d)$ has determinant $\pm 1$ (equivalently, ...

1
vote

1
answer

62
views

### Existence of some lattice path connecting all given lattice paths

My daily work concerns analysis on metric spaces and some time ago it turned out that the problem I am dealing with boils down to a certain combinatorial problem. I've checked a lot of examples and it ...

1
vote

0
answers

57
views

### Cohomology of cocompact lattices in hyperbolic spaces

I have a (maybe too naive) hope that cocompact torsion-free arithmetic lattices in hyperbolic spaces $X \neq \mathbb{H}_\mathbb{R}^2$ are uniquely determined by their cohomology with coefficients in $\...

2
votes

0
answers

182
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### Modular inverse computation - avoiding Euclidean algorithm

Modular inverse is known to be computable by Extended Euclidean algorithm which is the reaping the rewards of computing the GCD of two numbers or proving two numbers are coprime.
If we already know ...

0
votes

0
answers

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### On a number theory constraint

Let $a,b$ be odd coprime positive integers and let $u,v$ be reals such that:
$au+bv=1$
$a^ku+b^lv=c\in\mathbb Z$
holds at some fixed $k,l\in\mathbb Z_{\geq1}$ with $k+l\geq2$ with $|c|<\max(|a^{...

4
votes

1
answer

255
views

### The function $f(g):=\sum_{\gamma \in \text{SL}(2,\mathbb Z)}\frac{1}{\|g\gamma \|^4}$ for $g\in \text{SL}(2,\mathbb R)$

For $g\in \text{SL}(2,\mathbb R)$ and the Hilbert-Schmidt norm $\|\cdot\|$ (square root of sum of squares), define
$$f(g):=\sum_{\gamma \in \text{SL}(2,\mathbb Z)}\frac{1}{\|g\gamma \|^4}.$$
It is ...

1
vote

2
answers

122
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### Distribution of "good" and "bad" basis in lattice families?

I'm trying to learn more about lattice based cryptosystems.
One of the fundamental ideas behind lattice based cryptosystems is that there can be many equivalent basis for a single lattice.
Formally, ...

2
votes

0
answers

99
views

### Non-uniform lattices and parabolic subgroups in Lie groups

Let $G$ be a semisimple connected Lie group and let $\Lambda < G$ be a non-uniform irreducible lattice.
How does it follows that there exists a minimal parabolic subgroup $P$ of $G$ such that the ...

4
votes

1
answer

104
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### Inheritance of arithmeticity properties in orbifold strata

Suppose $M = K\backslash G/\Gamma$ is a quotient of a symmetric space by a lattice. I don't know all of the proper adjectives to apply here (e.g. what should be said about $G$ and so on), but I wouldn'...

3
votes

0
answers

68
views

### Torsion in the first cohomology of a lattice in a semisimple Lie group

Let $\Gamma$ be a cocompact lattice in a complex semisimple Lie group $G$ of dimension $n$. Let $M$ be a $\mathbb{Z}\Gamma$-module, finitely generated as an abelian group (let $r$ be the minimal ...

14
votes

1
answer

501
views

### How do we know there are no more Deligne–Mostow/Thurston lattices?

In the context of hypergeometric functions, Deligne and Mostow enumerated several lattices in complex hyperbolic space/the rank 1 Lie group $\operatorname{PU}(1,n)$ (see [1] and [2]). Thurston used ...

0
votes

1
answer

108
views

### How can I find the hyperplane passing through a 600-cell

I have a 600-cell, whose coordinates are given by
$$\begin{array}{ccc}
\text{8 vertices} & \left(0,0,0,\pm1\right) & \text{all permutations,}\\
\text{16 vertices} & \frac{1}{2}\left(\pm1,\...

10
votes

0
answers

195
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### Are topological theta series (taking values in tmf(N)) of lattices good for anything?

I'm going to start with Mike Hopkins' great survey article in the ICM on topological modular forms (https://arxiv.org/abs/math/0212397). In it, he outlines a construction, for even unimodular lattices,...

3
votes

0
answers

42
views

### maximizing number of lattice points with bounded diameter

Suppose I have a lattice $L$ that's just $\mathbb{Z}^k$ but scaled in every coordinate by some (potentially different) real numbers. Now I want to construct a finite set of lattice points $S \subset L$...

3
votes

0
answers

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### Can you find a Darboux basis for any skew integral form on a full-rank lattice in $ℂ^n$ so that the first $n$ vectors are $ℂ$-linearly independent?

Any skew bilinear form $\omega$ on $\mathbb{Z}^{2n}$ can be brought into the form
\begin{equation}
\begin{pmatrix}
0 & \Delta \\
-\Delta & 0
\end{pmatrix},
\quad
...

7
votes

0
answers

204
views

### K3 surfaces with no −2 curves

I seem to remember that a K3 surface with big Picard rank always
has smooth rational curves.
This question is equivalent to the following question about integral quadratic lattices. Let us call a ...

1
vote

1
answer

124
views

### What lattices beyond the laminated lattices (particularly in $\le 24D$) belong to a slightly expanded category that includes "descendants" of Λ13_mid?

This question is a copy of one I asked in the Math StackExchange forum a few days ago. I don't know if it qualifies as a research-level question, but it may be something beyond most people on the ...

2
votes

0
answers

80
views

### To show $\{(x,y) \in \mathbb Q^{\geq 0} \times \mathbb Q^{\geq 0}~:~ mn+1 \mid m^x+n^y \}$ is subset of the lattice $\{\vec u+i \vec v+j \vec w\}$?

I am writing two definitions, the $1$st one is a cover in some sense while the $2$nd one is a lattice:
Definition 1: If $m,n$ are integers bigger than $1$, then the set $$A=\{(x,y) \in \mathbb Q^{\geq ...

5
votes

1
answer

280
views

### What is the relationship between the Leech lattice and Dedekind eta function?

Like this old question, A conceptual proof of Jacobi's product formula for $\Delta$ ?, I am asking again for a conceptual proof of Jacobi's miraculous product formula for $\Delta$ (the unique ...

6
votes

1
answer

235
views

### Lattices in $p$-adic groups

What are the examples of lattices in $\operatorname{SL}_n(\mathbb{Q}_p)$ with $n\geq 3$ or in other semisimple $p$-adic groups of higher rank?
It is known $\operatorname{SO}_n(\mathbb{Z}[1/p])$ is a ...

2
votes

0
answers

142
views

### Mistake in Rogers' paper: "number of lattice points in a set" for the case $n=2$?

Let $f:\mathbb R^n\to \mathbb R$ be a nonnegative Borel measurable function, and
let $f^*$ be the function obtained from $f$ by spherical symmetrization (see Rogers' paper: number of lattice points in ...

11
votes

2
answers

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views

### Is there a contractible hyperbolic 3-orbifold of finite volume?

Let $\mathbb{H}^3:=\operatorname{SO}(3,1)/\operatorname{O(3)}$.
Is there a lattice $\Gamma$ in $\operatorname{SO}(3,1)$ such that
\begin{equation}
X:=\mathbb{H}^3/\Gamma
\end{equation}
is contractible?...

1
vote

0
answers

49
views

### Second moment version of the multiple-sum Rogers integration formula

I know the following theorems due to Rogers. Let $X$ denote the space of $n$-dimensional unimodualar lattices in $\mathbb R^n$, equipped with the canonical Haar measure.
Theorem 1(Siegel-Rogers). Let ...

5
votes

2
answers

208
views

### Is it still not known whether the construction of shortest nonzero vector of a lattice w.r.t. $l^2$-norm is NP-hard?

It was shown in
P. van Emde Boas, Another NP-complete partition problem and the complexity of computing short
vectors in a lattice
that the construction of a shortest nonzero vector of a Euclidean ...

3
votes

1
answer

137
views

### Entire function with almost periodic boundary condition?

Let $v_1 =\lambda_1 \zeta_1$ and $v_2 = \lambda_2 \zeta_2$ with $\zeta_1 = \frac{4\pi i\omega}{3}$ and $\zeta_2 = \frac{4\pi i\omega^2}{3}$ where $\omega = e^{2\pi i/3}$ is the third root of unity and ...

1
vote

0
answers

75
views

### Lattice packing

Let $\Lambda$ be a lattice in $R^n$ and $R>0$ a real number.
Consider the number $N$ of points in $\Lambda$ of norm less than $R$. Let $R$ goes to infinity. What can be said about the asymptotic ...

4
votes

2
answers

134
views

### How large is the set of unimodular lattices whose sucesssive minima cannot be attained by a basis of lattice?

Recall that the $i$-th successive minimum of $L\in \mathcal L$ (space of full rank lattices in $\mathbb R^d$), denoted $\lambda_i(L)$ is the infimum of the radii of the balls containing $i$-linearly ...

2
votes

1
answer

165
views

### Proof of generalized Siegel's mean value formula in geometry of numbers

Let $\mu$ be the Haar measure defined on the space of unimodular lattices, identified with $\text{SL}(d,\mathbb R)/\text{SL}(d,\mathbb Z)$.
The classical Siegel's formula in geometry of numbers states ...

1
vote

0
answers

123
views

### What is the lattice of the field $\mathbb Q_p(\sqrt[p^5-1]{p^2})$?

Let $p$ be odd prime and $\mathbb Q_p$ be the $p$-adic field. Consider the field extension $K=\mathbb Q_p(\sqrt[(p^5-1)]{p^2})$ of $\mathbb Q_p$ of degree $\frac{p^5-1}{2}$.
My question:
I want see ...

5
votes

1
answer

148
views

### Equivalence of quadratic forms over $p$-adic integers vs over localisation at $p$

To discern whether two integral quadratic forms are equivalent over the $p$-adic integers, one can compute a Jordan decomposition at $p$ and read off some invariants. Restricting to $p\ne2$ for ...

3
votes

0
answers

157
views

### Improvements to Minkowski's second theorem

Let $L$ be a (full rank) lattice in $\mathbb{R}^t$ and let $K$ be a convex body. Minkowski's second theorem states that
$$
\frac{2^t}{t!} \det(L) \leq \lambda_1 \cdot \ldots \cdot \lambda_t \text{Vol}(...

3
votes

1
answer

149
views

### Successive minima and the basis of lattice

I am able to prove the following two propositions: Recall that the $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the balls containing $i$-...

0
votes

0
answers

42
views

### Extension of primitive set of vectors and reduction theory

Let $\Lambda$ be a unimodular lattice in $\mathbb R^d$ (unimodularity is not really necessary here but just for convenience) and let $B$ be a ball centered at the origin that contains $(k+1)$-many $\...

1
vote

0
answers

47
views

### Finding a particular kind of basis of subgroup of a lattice generated by non-negative part

For $\mathbf v=(v_1,\ldots,v_n)\in \mathbb Z^n$, let $\operatorname{supp}(\mathbf v):=\{j: v_j \ne 0\}$. For a subset $X$ of $\mathbb Z^n$, define $\operatorname{supp}(X):=\bigcup_{\mathbf v \in X} \...

2
votes

0
answers

113
views

### Lattice relations and isogenous elliptic curves

Consider two (primitive) elements $\pi_{i} \in \mathbb{C}$, such that $\pi_{1} = M \pi_{2}$ for $M \in \mathcal{S}_{m}$ with $$\mathcal{S}_{m}:=\Big\{\begin{pmatrix}
A & B \\
0 & D
\end{...

1
vote

0
answers

35
views

### Barnes-Wall lattices’ contact polytopes

The contact polytopes of the Barnes-Wall lattices in 1, 2, 4, and 8 dimensions are all uniform polytopes. Is this true in any higher number of dimensions?

4
votes

0
answers

89
views

### Lattice reduction of basis with non-integer coefficients

Suppose I have an ordered basis $\{b_1, \dots, b_n\}$ of a lattice in $\mathbb{R}^n$, but I do not assume that $b_i \in \mathbb{Z}^n$ for all $1 \leq i \leq n$.
I would like to perform lattice ...

2
votes

0
answers

68
views

### Sums over lattice points in homogeneously expanding domains

In his book Algebraic Number Theory (2nd ed., Thm 2 in p.128), Lang proves the following (well-known) auxiliary result. Let $D\subset\mathbb{R}^N$ with $(N-1)$-Lipschitz parametrizable boundary. Let $...

1
vote

1
answer

83
views

### A lattice in $ \operatorname{SL}_n $ is Ad-irreducible

$\DeclareMathOperator\SL{SL}$Let $ G $ be a noncompact simple Lie group. For example $ \SL_n $. Let $ \Gamma $ be a lattice in $ G $. Consider the action of $ \Gamma $ on the Lie algebra of $ G $ by ...

2
votes

0
answers

84
views

### Find the closest point of a lattice $\Lambda$ given the closest point for its union of cosets $\bigcup_i ({\bf r}_i+\Lambda)$

Suppose we have an $n$-dimensional lattice $\Lambda$, and a set of vectors ${\bf r}_i$, we can construct a union of cosets of $\Lambda$, denoted as $L$, as
$$
L \equiv \bigcup_i ({\bf r}_i+\Lambda)
$$
...

0
votes

0
answers

43
views

### Minimal number of half-spaces needed to cover a lattice

Let $\Lambda$ be a lattice in $\mathbb{R}^n$, and $w\in\mathbb{R}^n-\Lambda$. A set $S\subseteq\Lambda$ of lattice points will be called a set of waypoints for $\Lambda$ from $w$ if it has any of the ...

6
votes

0
answers

138
views

### Why should Serre's conjecture on congruence subgroup property hold?

There seem to be several related properties of an algebraic group, exhibiting the dichotomy between rank 1 and rank $\ge2$.
Whether a lattice in the group satisfies the congruence subgroup property,
...

2
votes

2
answers

111
views

### Successive minima of a lattice and projection along the the shortest nonzero vector

Let $\mathcal L$ be the space of lattices in $\mathbb R^d$. The $i$-th successive minimum of $L\in \mathcal L$, denoted $\lambda_i(L)$ is the infimum of the radii of the balls containing $i$-linearly ...

10
votes

0
answers

291
views

### Kissing the Monster, or $196,560$ vs. $196,883$

The $D = 24$ kissing number is $196,560$, and the dimension of the smallest non-trivial complex representation of the Monster group is $196,883$. These two numbers are nearly but not quite equal, and ...