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
8
votes
1
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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 ...
- 161
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 ...
- 51
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 ...
- 51
0
votes
0
answers
19
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).
...
- 295
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 &...
- 307
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$.
...
- 141
2
votes
0
answers
77
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, ...
- 55
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 ...
- 117
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 $\...
- 111
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 ...
- 13.2k
0
votes
0
answers
71
views
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^{...
- 13.2k
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}.$$
It is ...
- 385
1
vote
2
answers
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Distribution of "good" and "bad" basis in lattice families?
I'm trying to learn more about lattice based cryptosystems.
One of the fundamental ideas behind lattice based cryptosystems is that there can be many equivalent basis for a single lattice.
Formally, ...
- 51
2
votes
0
answers
99
views
Non-uniform lattices and parabolic subgroups in Lie groups
Let $G$ be a semisimple connected Lie group and let $\Lambda < G$ be a non-uniform irreducible lattice.
How does it follows that there exists a minimal parabolic subgroup $P$ of $G$ such that the ...
- 419
4
votes
1
answer
104
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Inheritance of arithmeticity properties in orbifold strata
Suppose $M = K\backslash G/\Gamma$ is a quotient of a symmetric space by a lattice. I don't know all of the proper adjectives to apply here (e.g. what should be said about $G$ and so on), but I wouldn'...
- 1,045
3
votes
0
answers
68
views
Torsion in the first cohomology of a lattice in a semisimple Lie group
Let $\Gamma$ be a cocompact lattice in a complex semisimple Lie group $G$ of dimension $n$. Let $M$ be a $\mathbb{Z}\Gamma$-module, finitely generated as an abelian group (let $r$ be the minimal ...
- 161
14
votes
1
answer
501
views
How do we know there are no more Deligne–Mostow/Thurston lattices?
In the context of hypergeometric functions, Deligne and Mostow enumerated several lattices in complex hyperbolic space/the rank 1 Lie group $\operatorname{PU}(1,n)$ (see [1] and [2]). Thurston used ...
- 1,045
0
votes
1
answer
108
views
How can I find the hyperplane passing through a 600-cell
I have a 600-cell, whose coordinates are given by
$$\begin{array}{ccc}
\text{8 vertices} & \left(0,0,0,\pm1\right) & \text{all permutations,}\\
\text{16 vertices} & \frac{1}{2}\left(\pm1,\...
- 295
10
votes
0
answers
195
views
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,...
- 333
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 $S \subset L$...
- 480
3
votes
0
answers
60
views
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
...
- 1,081
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 ...
- 8,925
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 ...
- 247
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 ...
- 419
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 ...
- 347
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 ...
- 385
11
votes
2
answers
1k
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?...
- 423
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 ...
- 385
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 ...
- 385
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 ...
- 53
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 ...
- 237
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 ...
- 385
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 ...
- 385
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 ...
- 247
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 ...
- 323
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}(...
- 432
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 $i$-...
- 385
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 $\...
- 385
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} \...
- 171
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 $$\mathcal{S}_{m}:=\Big\{\begin{pmatrix}
A & B \\
0 & D
\end{...
- 61
1
vote
0
answers
35
views
Barnes-Wall lattices’ contact polytopes
The contact polytopes of the Barnes-Wall lattices in 1, 2, 4, and 8 dimensions are all uniform polytopes. Is this true in any higher number of dimensions?
- 2,415
4
votes
0
answers
89
views
Lattice reduction of basis with non-integer coefficients
Suppose I have an ordered basis $\{b_1, \dots, b_n\}$ of a lattice in $\mathbb{R}^n$, but I do not assume that $b_i \in \mathbb{Z}^n$ for all $1 \leq i \leq n$.
I would like to perform lattice ...
- 554
2
votes
0
answers
68
views
Sums over lattice points in homogeneously expanding domains
In his book Algebraic Number Theory (2nd ed., Thm 2 in p.128), Lang proves the following (well-known) auxiliary result. Let $D\subset\mathbb{R}^N$ with $(N-1)$-Lipschitz parametrizable boundary. Let $...
- 2,949
1
vote
1
answer
83
views
A lattice in $ \operatorname{SL}_n $ is Ad-irreducible
$\DeclareMathOperator\SL{SL}$Let $ G $ be a noncompact simple Lie group. For example $ \SL_n $. Let $ \Gamma $ be a lattice in $ G $. Consider the action of $ \Gamma $ on the Lie algebra of $ G $ by ...
- 2,877
2
votes
0
answers
84
views
Find the closest point of a lattice $\Lambda$ given the closest point for its union of cosets $\bigcup_i ({\bf r}_i+\Lambda)$
Suppose we have an $n$-dimensional lattice $\Lambda$, and a set of vectors ${\bf r}_i$, we can construct a union of cosets of $\Lambda$, denoted as $L$, as
$$
L \equiv \bigcup_i ({\bf r}_i+\Lambda)
$$
...
- 163
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 ...
- 486
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,
...
- 952
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 ...
- 385
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 ...
- 209