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])

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Is there an Ehrhart polynomial for Gaussian integers

Let $N$ be a positive integer and let $P \subset \mathbb{C}$ be a polygon whose vertices are of the form $(a_1+b_1 i)/N$, $(a_2+b_2 i)/N$, ..., $(a_r+b_r i)/N$, with $a_j + b_j i$ being various ...
David E Speyer's user avatar
12 votes
0 answers
285 views

Who first showed that $SL(n,O_K)$ is a lattice for a number ring $O_K$?

Let $O_K$ be the ring of integers in an algebraic number field $K$. Assume that $K$ has $r$ real embeddings and $s$ pairs of complex conjugate complex embeddings. There is then an injective ...
Melanie's user avatar
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12 votes
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547 views

Connes & Marcolli: Q-lattices generalize Conway's "Understanding groups like $\Gamma_0(N)$"

Has anyone generalized Conway's description of Hecke operators on lattices to the Q-lattices of Connes & Marcolli ? Light may well be shone on moonshine thus.
John McKay's user avatar
11 votes
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158 views

Characterization of certain 4-dimensional lattices

Let $\Lambda \subset {\bf Q}^4$ be a lattice, i.e., $\Lambda$ is a free abelian group and $\Lambda \otimes {\bf Q} = {\bf Q^4}$. The determinants of those dilation-rotations (i.e. linear maps of ${\bf ...
Jens Reinhold's user avatar
11 votes
0 answers
215 views

Representation of the space of lattices in $\Bbb R^n$

The space of 2D lattices in $\Bbb R^2$ can be represented with the two Eisenstein series $G_4$ and $G_6$. Each lattice uniquely maps to a point in $\Bbb C^2$ using these two invariants, and the points ...
Mike Battaglia's user avatar
10 votes
0 answers
206 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,...
Mike's user avatar
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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 ...
Harry Wilson's user avatar
10 votes
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1k views

Number of rectangles in an n-by-n grid of points

I am seeking a formula for the number of rectangles with vertices belonging to an $n \times n$ square grid of points. This is sequence A085582 in the OEIS. Note that the grid in question has $n^2$ ...
Dave R's user avatar
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10 votes
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Bound on the number of lattice points in d-dimensional ball

The following paper states that the number of lattice points in a $d$-dimensional ball of radius $R$ is $V_d R^d + O(R^\alpha)$ where $\alpha = d - 2$ and $V_d$ is the volume of the unit $d$-...
Guy's user avatar
  • 201
9 votes
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Conway big picture for congruence subgroups of $\mathrm{SL}_3(\mathbb{Z})$

I saw in Conway’s paper "Understanding groups like $\Gamma_0(N)$" that the so-called Big Picture can give simple interpretations for important objects in number theory, such as Hecke ...
Radu T's user avatar
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A lattice with Monster group symmetries

The book Mathematical Evolutions contains the following excerpt: A last, famous, example is the following. It is known that in the space of one hundred and ninety six thousand eight hundred and ...
Adam P. Goucher's user avatar
9 votes
0 answers
354 views

How to count integer lattice points close to a subspace of $\mathbb R^n$?

Consider $m$ linearly independent vectors in $n$-dimensional Euclidean space, $v_1,...,v_m \in \mathbb R^n$ where $1\leq m<n$, and let $U := {\rm span}(v_1,...,v_m)$ denote the $m$-dimensional ...
Dierk Bormann's user avatar
9 votes
0 answers
132 views

Is there a reason nice coset representatives exist for Leech or E_8 lattice modulo 2?

Let $\Lambda$ be the Leech lattice. There is a nice set of coset representatives for $\Lambda/2 \Lambda$ given by short vectors [Conway and Sloane, Ch. 10, Theorem 28 or Ch. 23, Theorem 3]. The proof ...
Tathagata Basak's user avatar
8 votes
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251 views

Positive integer solutions of $ab+ac+ad+bc+bd+cd=n$

Consider a quadratic form $Q(a,b,c,d)=ab+ac+ad+bc+bd+cd$ on $\mathbb Z^4$. For some reason I am interested in the number of solutions $(a,b,c,d)\in\mathbb Z_{> 0}^4$ of $Q(a,b,c,d)=n$ as a function ...
Nikita Kalinin's user avatar
8 votes
0 answers
306 views

Minkowski's convex body theorem for ellipsoids

Minkowski's theorem states that if $K\subseteq\mathbb{Z}^n$ is a convex compact set, $K=-K$, and $\mathrm{volume}(K)\geq 2^n$, then $K$ contains a nonzero integral vector. Can this bound be improved ...
Marcel Celaya's user avatar
7 votes
0 answers
236 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 ...
Misha Verbitsky's user avatar
7 votes
0 answers
211 views

How to check two matrices for similitude over $\mathbb{Z}$?

General question. Let $A$ and $B$ be two $n\times n$-matrices over $\mathbb{Z}$. How do I algorithmically check whether $A$ and $B$ are similar (i.e., conjugate in the ring $\mathbb{Z}^{n\times n}$)? ...
darij grinberg's user avatar
7 votes
0 answers
116 views

Theta Function Associated to Kummer Lattice

This is something which I feel must be out in the literature somewhere, but I have been unable to find anything. If we let $\text{Km}(A)$ be the Kummer $K3$ surface associated to an abelian surface $A$...
Benighted's user avatar
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7 votes
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intuition for lattices in p-adic (or other non-Archimedean) vector spaces?

I could use some help to jumpstart my intuition for lattices in vector spaces over non-Archimedean fields, like $\mathbb{Q}_p$ and $\mathbb{F}_q((t))$. I have some intuition for $\mathbb{Z}$-lattices ...
PrimeRibeyeDeal's user avatar
7 votes
1 answer
499 views

Counting points on the intersection of a box and a lattice

Let $A:\mathbb{Z}^n\to \mathbb{Z}^n$ be non-singular. Consider a box $B=[0,N_1]\times [0,N_2] \times \dotsc \times [0,N_n]$. Let $p_1,\dotsc,p_n$ be primes (distinct, if you wish) and let $L = p_1\...
H A Helfgott's user avatar
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7 votes
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160 views

Constructive proof of Swan theorem

Let $M$ be an $S_n$-lattice (so it is free as an abelian group), and assume that $M$ is projective (i.e. direct summand of some $\mathbb Z[S_n]^m$). A theorem of Swan implies that $M$ is stably ...
S. du Val's user avatar
  • 161
7 votes
0 answers
171 views

Subgroups of $\mathbb{Z}^{n}$ with rotational symmetries

Schmidt (https://projecteuclid.org/euclid.dmj/1077377618) showed that the number of $m$-dimensional subgroups of $\mathbb{Z}^{n}$ of covolume $\leq X$ is $$c_{1}\left(m,n\right)X^{n}+O\left(X^{n-\...
Tal H's user avatar
  • 273
7 votes
0 answers
253 views

Question on some coverings of the euclidean space

Let $L$ be a maximal integral lattice in the euclidean $(\mathbf R^{8m},q)$ (thus the associated bilinear form $b(u,v)=q(u+v)-q(u)-q(v)$, once restricted to $L$, takes values in $2\mathbf Z$ and has ...
few_reps's user avatar
  • 1,980
7 votes
0 answers
190 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, $b(x,...
eins6180's user avatar
  • 1,302
6 votes
0 answers
227 views

What can lattices tell us about lattices?

A general group-theoretic lattice is usually defined as something like A discrete subgroup $\Gamma$ of a locally compact group $G$ is a lattice if the quotient $G/\Gamma$ carries a $G$-invariant ...
Mark Schultz-Wu's user avatar
6 votes
0 answers
172 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, ...
GTA's user avatar
  • 1,014
6 votes
0 answers
212 views

Divisor bound for $r_2$ off the origin

If $r_2(n)$ denotes the number of integer solutions to $a^2+b^2=n$, we have the "divisor bound" $r_2(n) = O(n^{\epsilon})$ for any $\epsilon>0$. Another way to state this is that the number of ...
Rodrigo's user avatar
  • 1,235
6 votes
0 answers
206 views

Counting lattice points in adelic spaces

Let $\mathbb{A}$ denote the ring of adeles of $\mathbb{Q}$, let $\mu$ be the Haar measure of $\mathbb{A}$, and let $\|\cdot\|_{\infty}$ denote the sup-norm of the components in the Archimedean ...
Maurizio Moreschi's user avatar
6 votes
0 answers
273 views

Formula for Schinzel circle with minimum radius

Schinzel's Theorem states that there are a set of circles with a given number of integer points on the circumference of the circle. The theorem includes the equation for an instance of circle given $n$...
Simple Mistake's user avatar
6 votes
0 answers
174 views

Shortest vector problem with a null vector constraint

Take $\mathbb{Z}^n$ equipped with two symmetric bilinear forms, one positive-definite $(\cdot,\cdot)_A : \mathbb{Z}^n \times \mathbb{Z}^n \to \mathbb{R}$ and one indefinite $(\cdot,\cdot)_J : \mathbb{...
Chaitanya Murthy's user avatar
6 votes
0 answers
169 views

Root system inside the indefinite even unimodular lattice $II_{10,2}$

I apologize for asking questions that seem likely to be answered in Conway & Sloane's "Sphere Packings, Lattices, and Groups" if I knew where to look. Let $L$ be the unique* even unimodular ...
Allen Knutson's user avatar
6 votes
0 answers
266 views

Bound on the determinant of a quadratic form restricted to a subspace

Let $Q\colon \mathbb{Z}^{n}\oplus\mathbb{Z}^m\to\mathbb{R}$ be a real quadratic form, which we denote $Q(x,y)$, $x\in\mathbb{Z}^n$, $y\in\mathbb{Z}^m$. Suppose: The minimum of $Q(x,y)$ as $y$ varies ...
Yoav Kallus's user avatar
  • 5,926
6 votes
0 answers
547 views

Is this property a new large cardinal notion?

Given a cardinal $\kappa$, $\kappa$-complete lattices are lattices that have joins and meets of less than $\kappa$ elements (in particular they are bounded). In what follows we shall restrict to the ...
godelian's user avatar
  • 5,632
5 votes
0 answers
139 views

Is fundamental group of a finite volume, negatively curved, cusped manifold a non-uniform lattice?

$\DeclareMathOperator\Mob{Mob}$Some background: (1) A Riemannian manifold $M$ is pinched negatively curved if there is a constant $\tau<\kappa<0$ such that all the sectional curvatures are ...
Yanlong Hao's user avatar
5 votes
0 answers
277 views

Matrix groups with two generators

Given two matrices $A,B\in{\rm{SL}}_2(\Bbb{R})$, is there any criterion guaranteeing that the subgroup they generate is discrete? What if one puts restrictions on $A,B$ e.g. they are both elliptic? ...
KhashF's user avatar
  • 2,588
5 votes
0 answers
149 views

Lattices of minimal covolume in $\operatorname{SL}_2(\mathbb{R}) \times \operatorname{SL}_2(\mathbb{R})$

What are the (uniform/non-uniform) irreducible lattices of minimal (or even small) covolume in $\operatorname{SL}_2(\mathbb{R}) \times \operatorname{SL}_2(\mathbb{R})$? Context: Such a lattice will ...
Stefan Witzel's user avatar
5 votes
0 answers
176 views

Examples of non-uniform lattices in products of trees

Consider a product of two locally-finite, infinite, unimodular trees $X=T_1\times T_2$. Assume that both ${\rm Aut}(T_1)$ and ${\rm Aut}(T_2)$ are not discrete. So as a vague general question, what ...
Sam Hughes's user avatar
5 votes
0 answers
443 views

Lattices in Lie groups

In the literature, people seem to predominantly look at lattices in nilpotent or reductive groups. Is there a result that gives a general description of a lattice in an arbitrary Lie group? Something ...
user avatar
5 votes
0 answers
213 views

Isomorphism classes of lattices

Suppose we have a $4 \times 6$ matrix $A$ of rank $4$ whose entries are rational numbers. Define $$ V = \{x \in \mathbb R^6 \mid A \cdot x = 0\} $$ and $$ \Lambda = \{x \in \mathbb Z^6 \mid A \cdot ...
Ramin's user avatar
  • 1,362
5 votes
0 answers
128 views

Averaging number of lattice points in a box over a family of lattices

Consider the diophantine equation $$ x_1y_1^3 + \dots + x_s y_s^3 = 0. $$ For fixed $\mathbf{y}$ with coprime coordinates this is a $s-1$ dimensional lattice $\Lambda(\mathbf{y})$. Let $N(X)$ denote ...
leithian's user avatar
  • 163
5 votes
0 answers
322 views

Selmer Group of number fields and Ideal lattices

Let $K$ be a totally real number field of degree $n$ and dicriminant $d$, in this article of F.Lemmermeyer the selmer group of $K$ is defined as $$\text{Sel}(K):=\{\alpha \in K^{\times}: (\alpha)=...
user avatar
5 votes
0 answers
104 views

How well can a rotation separate lattice vectors of equal norm in Z^d?

I'm interested in rotations $R$ that maximally separate integral lattice vectors of equal norm. This question is preliminary, and regards the scaling of those separations as norm goes to infinity. ...
Chaitanya Murthy's user avatar
5 votes
0 answers
742 views

Is this set empty?

Suppose we have two rank $n-1$ matrices in $\Bbb Z^{(n-1)\times n}$ given by $$C=\begin{bmatrix} c_{1}& -d_{1}& 0& 0&\dots 0& 0\\ 0& c_{2}& -d_{2}& 0&...
Turbo's user avatar
  • 13.6k
5 votes
0 answers
128 views

Lattice paths in polytopes

Let $P$ be a polytope in $\mathbb{R}^n$. Let $A_ix = b_i$ be the defining equations of its codimension $1$ faces. Is there an algorithm or some kind of criterion to decide if the lattice points inside ...
Alex's user avatar
  • 491
5 votes
0 answers
305 views

Are the homogeneous single chain subfactors, Dedekind?

Background: See here and there. Recall that a subfactor is Dedekind if all its intermediate subfactors are normal. A subfactor $(N \subset M)$ is Homogeneous Single Chain (HSC) if its lattice ...
Sebastien Palcoux's user avatar
5 votes
0 answers
136 views

Minimum of the product of linear forms over a lattice

In Chapter [IX.1] of Siegel's Lectures on the Geometry of Numbers it is shown that if we have $n$ linear forms $y_{j}=\sum_{k=1}^{n}{a_{jk}x_{k}},\quad j=1,\ldots,n$, with the coefficient matrix $(a_{...
Felix Goldberg's user avatar
5 votes
0 answers
315 views

Lattice points inside a (n-dimensional) tetrahedron

Hi, overflowers. I was interested in a sharp lower bound for the number of lattice points (say, integral lattice points) inside the tetrahedron defined by the coordinate hyperplanes and $x_1/a_1+...+...
Chema Tornero's user avatar
5 votes
0 answers
265 views

Automorphisms of Torsion Quadratic Forms

Let $L$ be an $n$-dimensional lattice with an integer valued quadratic form $q$. Fix a basis $e_i$ for $L$ and let $K_{ij} = \langle e_i, e_j \rangle$, where $\langle x,y \rangle = q(x+y) - q(x) - q(...
Lukasz Fidkowski's user avatar
5 votes
0 answers
202 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 ...
Will Jagy's user avatar
  • 25.3k
4 votes
0 answers
134 views

Vanishing of $\ell^2$-Betti numbers of $\mathrm{GL}(n,\mathbb{Z})$ for $n\geq 3$

$\DeclareMathOperator\GL{GL}$In a paper I read the following claim: By the work of Borel the $\ell^2$-Betti numbers of the cocompact lattices of $\GL(n,\mathbb{R})$ are known to all vanish when $n ≥ 3$...
Marcos's user avatar
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