# Dense lattices as Hermitian tensor products

@inproceedings{Coulangeon2013DenseLA, title={Dense lattices as Hermitian tensor products}, author={Renaud Coulangeon and Gabriele Nebe}, year={2013} }

Using the Hermitian tensor product description of the extremal even unimodular lattice of dimension 72 found by Nebe in 2010 we show its extremality with the methods from Coulangeons article in Acta Arith. 2000.

## 6 Citations

### On extremal even unimodular 72-dimensional lattices

- Mathematics, Computer ScienceMath. Comput.
- 2014

By computer search we show that the lattice Γ from [9] is the unique extremal even unimodular 72-dimensional lattices that can be constructed as proposed in [6].

### Golden lattices

- Mathematics
- 2012

Let θ := −1+ √ 5 2 be the golden ratio. A golden lattice is an even unimodular Z[θ]-lattice of which the Hilbert theta series is an extremal Hilbert modular form. We construct golden lattices from…

### J un 2 02 1 UNIRULEDNESS OF UNITARY SHIMURA VARIETIES ASSOCIATED WITH HERMITIAN FORMS OF SIGNATURES

- Mathematics
- 2021

Gritsenko-Hulek showed that some orthogonal Shimura varieties are uniruled. In particular, they have the Kodaira dimension ́8. In this paper, we apply the method of Gritsenko-Hulek to unitary…

### On extremal lattices in jump dimensions

- Mathematics
- 2012

Let (L,Q) be an even unimodular lattice, so L is a free Z-module of rank n, and Q : L → Z a positive definite regular integral quadratic form. Then L can be embedded into Euclidean n-space (R, (, ))…

### Uniruledness of unitary Shimura varieties associated with Hermitian forms of signatures $(1,4)$ and $(1,5)$

- Mathematics
- 2020

Gritsenko-Hulek showed that some orthogonal Shimura varieties are uniruled. In particular, they have Kodaira dimension $-\infty$. In this paper, we apply the method of Gritsenko-Hulek to unitary…

### A new framework for building and decoding group codes

- Computer ScienceArXiv
- 2020

This paper investigates the construction and the decoding of a remarkable set of lattices and codes viewed as group codes: We treat in a unified framework the Leech lattice and the Golay code in…

## References

SHOWING 1-10 OF 27 REFERENCES

### A Mordell inequality for lattices over maximal orders

- Mathematics
- 2008

In this paper we prove an analogue of Mordell's inequality for lattices in finite-dimensional complex or quaternionic Hermitian space that are modules over a maximal order in an imaginary quadratic…

### On positive definite hermitian forms

- Mathematics
- 1991

Kneser's method of constructing adjacent lattices will be used to determine class numbers of unimodular positive definite hermitian lattices of rank 2 and 3 over rings of integers in some imaginary…

### Some Cyclo-quaternionic Lattices☆

- Mathematics
- 1998

Abstract The faithful lattices of rank 2( p − 1) of the groups SL 2 ( p ) are described. For small primes p these and related lattices are investigated by computer. In particular a new extremal even…

### An even unimodular 72-dimensional lattice of minimum 8

- Mathematics
- 2010

Abstract An even unimodular 72-dimensional lattice Γ having minimum 8 is constructed as a tensor product of the Barnes lattice and the Leech lattice over the ring of integers in the imaginary…

### EXTREMAL LATTICES OF MINIMUM 8 RELATED TO THE MATHIEU GROUP M22

- Mathematics
- 1998

In this paper, we construct three new extremal lattices of minimum 8; one is 3-modular and of dimension 40, the two others are unimodular of dimension 80. They are strongly connected to the…

### Symmetric Bilinear Forms

- Mathematics
- 1973

I. Basic Concepts.- II. Symmetric Inner Product Spaces over Z.- III. Inner Product Spaces over a Field.- IV. Discrete Valuations and Dedekind Domains.- V. Some Examples.- Appendix 1. Quadratic…

### On the Extremality of an 80-Dimensional Lattice

- MathematicsANTS
- 2010

We show that a specific even unimodular lattice of dimension 80, first investigated by Schulze-Pillot and others, is extremal (i.e., the minimal nonzero norm is 8). This is the third known extremal…

### The Magma Algebra System I: The User Language

- Computer ScienceJ. Symb. Comput.
- 1997

The MAGMA language is presented, the design principles and theoretical background are outlined, and the constructors for structures, maps, and sets are outlined.