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Prof. Dr. Gabriele Nebe

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Lehrstuhl D für Mathematik, RWTH Aachen, Germany
Interests: integral and orthogonal representations of finite groups; lattices, spherical designs and modular forms, and their analogues in coding theory

Special Issue Information

Dear Colleagues,

Geometry of Numbers is a famous and classical area of mathematics founded by Hermann Minkowski to apply the theory of lattices in Euclidean space to important problems in algebraic number theory. Even in this classical orientation it is still a vivant area, as we see from the recent achievements by Curtis T. McMullen on the Minkowski conjecture and the work on Euclidean minima of number fields initiated by Eva Bayer-Fluckiger. But the topic has substantially grown, and the aim of the present Special Issue is to collect original research articles, as well as a few high level survey articles focusing on connections between lattices and number theory. Among the important areas are, from my personal perspective: Arakelov geometry, diophantine approximation, K-theory and the cohomology of arithmetic groups, and of course the classical topics such as lattices and modular forms.

Already in the actual theory of lattices in Euclidean spaces, there are quite a few very remarkable recent results, such as the proof that the maximum density of a sphere packing in dimension 8 resp. 24 is realized by the E₈-lattice, respectively the Leech lattice by Viazovska et al., and the discovery of new extremal even unimodular lattices in dimension 48 and 72. Also the counter-example to Woods conjecture given by Regev, Shapira, and Weiss and showing that McMullen’s approach to prove Minkowski’s conjecture fails in higher dimensions, will certainly stimulate the research in this area.

AMS-Classification include:

11E12 Quadratic forms over global rings and fields
11F11 Modular forms, one variable
11F75 Cohomology of arithmetic groups
11HXX Geometry of Numbers
13F07 Euclidean rings and generalizations
14G40 Arithmetic varieties and schemes; Arakelov theory
20C10 Integral representations of finite groups

Prof. Dr. Gabriele Nebe
Guest Editor

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Keywords

Lattices in Euclidean spaces
Sphere packing problem
Sphere covering problem
Automorphism groups of lattices
Connections to modular forms
Lattices with algebraic structure
Application of lattices in number theory
Arithmetic groups and Cohomology
Arakelov geometry
Hyperbolic lattices

Published Papers (3 papers)

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Research

407 KiB

Open AccessArticle

Multiplicative Structure and Hecke Rings of Generator Matrices for Codes over Quotient Rings of Euclidean Domains

by Hajime Matsui

Mathematics 2017, 5(4), 82; https://doi.org/10.3390/math5040082 - 15 Dec 2017

Cited by 3 | Viewed by 2806

Abstract

In this study, we consider codes over Euclidean domains modulo their ideals. In the first half of the study, we deal with arbitrary Euclidean domains. We show that the product of generator matrices of codes over the rings mod a and mod b [...] Read more.

a b

, i.e., this correspondence is onto. Moreover, we show that if a and b are coprime, then this correspondence is one-to-one, i.e., there exist unique codes over the rings mod a and mod b that produce any given code over the ring mod

a b

through the product of their generator matrices. In the second half of the study, we focus on the typical Euclidean domains such as the rational integer ring, one-variable polynomial rings, rings of Gaussian and Eisenstein integers, p-adic integer rings and rings of one-variable formal power series. We define the reduced generator matrices of codes over Euclidean domains modulo their ideals and show their uniqueness. Finally, we apply our theory of reduced generator matrices to the Hecke rings of matrices over these Euclidean domains. Full article

(This article belongs to the Special Issue Geometry of Numbers)

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457 KiB

Open AccessArticle

On Minimal Covolume Hyperbolic Lattices

by Ruth Kellerhals

Mathematics 2017, 5(3), 43; https://doi.org/10.3390/math5030043 - 22 Aug 2017

Cited by 2 | Viewed by 3878

Abstract

We study lattices with a non-compact fundamental domain of small volume in hyperbolic space

H^{n}

. First, we identify the arithmetic lattices in

{Isom}^{+} H^{n}

of minimal covolume for even n up to 18. Then, we discuss the related problem [...] Read more.

We study lattices with a non-compact fundamental domain of small volume in hyperbolic space

H^{n}

. First, we identify the arithmetic lattices in

{Isom}^{+} H^{n}

of minimal covolume for even n up to 18. Then, we discuss the related problem in higher odd dimensions and provide solutions for

n = 11

and

n = 13

in terms of the rotation subgroup of certain Coxeter pyramid groups found by Tumarkin. The results depend on the work of Belolipetsky and Emery, as well as on the Euler characteristic computation for hyperbolic Coxeter polyhedra with few facets by means of the program CoxIter developed by Guglielmetti. This work complements the survey about hyperbolic orbifolds of minimal volume. Full article

(This article belongs to the Special Issue Geometry of Numbers)

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Figure 1

269 KiB

Open AccessArticle

Lattices and Rational Points

by Evelina Viada

Mathematics 2017, 5(3), 36; https://doi.org/10.3390/math5030036 - 09 Jul 2017

Cited by 1 | Viewed by 3114

Abstract

In this article, we show how to use the first and second Minkowski Theorems and some Diophantine geometry to bound explicitly the height of the points of rank

N - 1

on transverse curves in

E^{N}

, where E is an elliptic [...] Read more.

In this article, we show how to use the first and second Minkowski Theorems and some Diophantine geometry to bound explicitly the height of the points of rank

N - 1

on transverse curves in

E^{N}

, where E is an elliptic curve without Complex Multiplication (CM). We then apply our result to give a method for finding the rational points on such curves, when E has

Q

-rank

\leq N - 1

. We also give some explicit examples. This result generalises from rank 1 to rank

N - 1

previous results of S. Checcoli, F. Veneziano and the author. Full article

(This article belongs to the Special Issue Geometry of Numbers)

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