Tensor products of hermitian lattices

Renaud Coulangeon

Acta Arithmetica (2000)

  • Volume: 92, Issue: 2, page 115-130
  • ISSN: 0065-1036

Abstract

top
1. Introduction. The properties of euclidean lattices with respect to tensor product have been studied in a series of papers by Kitaoka ([K, Chapter 7], [K1]). A rather natural problem which was investigated there, among others, was the determination of the short vectors in the tensor product L οtimes M of two euclidean lattices L and M. It was shown for instance that up to dimension 43 these short vectors are split, as one might hope. The present paper deals with a similar question for tensor products of hermitian lattices over imaginary quadratic fields or quaternion division algebras. The main motivation for this work is in connection with modular lattices, as defined by Quebbemann ([Q]), that is to say, even lattices that are similar to their dual. In [B-N] it is shown how tensor product over the ring of integers in an imaginary quadratic field can be used to shift from one level to another (the level of a modular lattice L is the square of the rate of the similarity mapping L* to L), and above all a construction of an 80-dimensional extremal unimodular lattice from a 20-dimensional 7-modular one by tensoring is given. It is thus of some interest to know a priori how short vectors behave under tensor product. In Section 2 we give the basic definitions and properties concerning hermitian lattices that are needed in the sequel. We establish in Section 3 a splitness criterion for minimal vectors (Corollary 3.4) based on a general lower bound (Proposition 3.2). Finally, Section 4 is devoted to examples; among others, we give an alternate proof of the extremality of Bachoc-Nebe's 80-dimensional lattice, and we give a new construction of the Barnes-Wall lattices.

How to cite

top

Renaud Coulangeon. "Tensor products of hermitian lattices." Acta Arithmetica 92.2 (2000): 115-130. <http://eudml.org/doc/207374>.

@article{RenaudCoulangeon2000,
abstract = { 1. Introduction. The properties of euclidean lattices with respect to tensor product have been studied in a series of papers by Kitaoka ([K, Chapter 7], [K1]). A rather natural problem which was investigated there, among others, was the determination of the short vectors in the tensor product L οtimes M of two euclidean lattices L and M. It was shown for instance that up to dimension 43 these short vectors are split, as one might hope. The present paper deals with a similar question for tensor products of hermitian lattices over imaginary quadratic fields or quaternion division algebras. The main motivation for this work is in connection with modular lattices, as defined by Quebbemann ([Q]), that is to say, even lattices that are similar to their dual. In [B-N] it is shown how tensor product over the ring of integers in an imaginary quadratic field can be used to shift from one level to another (the level of a modular lattice L is the square of the rate of the similarity mapping L* to L), and above all a construction of an 80-dimensional extremal unimodular lattice from a 20-dimensional 7-modular one by tensoring is given. It is thus of some interest to know a priori how short vectors behave under tensor product. In Section 2 we give the basic definitions and properties concerning hermitian lattices that are needed in the sequel. We establish in Section 3 a splitness criterion for minimal vectors (Corollary 3.4) based on a general lower bound (Proposition 3.2). Finally, Section 4 is devoted to examples; among others, we give an alternate proof of the extremality of Bachoc-Nebe's 80-dimensional lattice, and we give a new construction of the Barnes-Wall lattices. },
author = {Renaud Coulangeon},
journal = {Acta Arithmetica},
keywords = {Hermitian lattice; tensor product of lattices; Barnes-Wall lattices},
language = {eng},
number = {2},
pages = {115-130},
title = {Tensor products of hermitian lattices},
url = {http://eudml.org/doc/207374},
volume = {92},
year = {2000},
}

TY - JOUR
AU - Renaud Coulangeon
TI - Tensor products of hermitian lattices
JO - Acta Arithmetica
PY - 2000
VL - 92
IS - 2
SP - 115
EP - 130
AB - 1. Introduction. The properties of euclidean lattices with respect to tensor product have been studied in a series of papers by Kitaoka ([K, Chapter 7], [K1]). A rather natural problem which was investigated there, among others, was the determination of the short vectors in the tensor product L οtimes M of two euclidean lattices L and M. It was shown for instance that up to dimension 43 these short vectors are split, as one might hope. The present paper deals with a similar question for tensor products of hermitian lattices over imaginary quadratic fields or quaternion division algebras. The main motivation for this work is in connection with modular lattices, as defined by Quebbemann ([Q]), that is to say, even lattices that are similar to their dual. In [B-N] it is shown how tensor product over the ring of integers in an imaginary quadratic field can be used to shift from one level to another (the level of a modular lattice L is the square of the rate of the similarity mapping L* to L), and above all a construction of an 80-dimensional extremal unimodular lattice from a 20-dimensional 7-modular one by tensoring is given. It is thus of some interest to know a priori how short vectors behave under tensor product. In Section 2 we give the basic definitions and properties concerning hermitian lattices that are needed in the sequel. We establish in Section 3 a splitness criterion for minimal vectors (Corollary 3.4) based on a general lower bound (Proposition 3.2). Finally, Section 4 is devoted to examples; among others, we give an alternate proof of the extremality of Bachoc-Nebe's 80-dimensional lattice, and we give a new construction of the Barnes-Wall lattices.
LA - eng
KW - Hermitian lattice; tensor product of lattices; Barnes-Wall lattices
UR - http://eudml.org/doc/207374
ER -

References

top
  1. [ATLAS] J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, Atlas of Finite Groups, Oxford Univ. Press, Oxford, 1985. Zbl0568.20001
  2. [B] C. Bachoc, Applications of coding theory to the construction of modular lattices, J. Combin. Theory Ser. A 78 (1997), 92-119. Zbl0876.94053
  3. [B-N] C. Bachoc and G. Nebe, Extremal lattices of minimum 8 related to the Mathieu group M 2 2 , J. Reine Angew. Math., to appear. 
  4. [B-W] E. S. Barnes and G. E. Wall, Some extreme forms defined in terms of Abelian groups, J. Austral. Math. Soc. 1 (1959), 47-63. Zbl0109.03304
  5. [Bou] N. Bourbaki, Algèbre, Hermann, Paris, 1970. 
  6. [C-S] J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, 3rd ed., Grundlehren Math. Wiss. 290, Springer, New York, 1999. 
  7. [H] D. W. Hoffmann, On positive definite Hermitian forms, Manuscripta Math. 71 (1991), 399-429. Zbl0729.11020
  8. [K] Y. Kitaoka, Arithmetic of Quadratic Forms, Cambridge Univ. Press, 1993. Zbl0785.11021
  9. [K1] Y. Kitaoka, Scalar extension of quadratic lattices II, Nagoya Math. J. 67 (1977), 159-164. Zbl0361.10020
  10. [M] J. Martinet, Les Réseaux Parfaits des Espaces Euclidiens, Masson, Paris, 1996. 
  11. [M-H] J. Milnor and J. D. Husemoller, Symmetric Bilinear Forms, Ergeb. Math. Grenzgeb. 73, Springer, New York, 1973. Zbl0292.10016
  12. [N-P] G. Nebe and W. Plesken, Finite rational matrix groups, Mem. Amer. Math. Soc. 556 (1995). Zbl0837.20056
  13. [Q] H.-G. Quebbemann, Modular lattices in euclidean spaces, J. Number Theory 54 (1995), 190-202. Zbl0874.11038
  14. [R] I. Reiner, Maximal Orders, Academic Press, London, 1975. 
  15. [Sc] A. Schiemann, private communication. 
  16. [Se] J.-P. Serre, Corps Locaux, Hermann, Paris, 1968. 
  17. [St] N. W. Stoltzfus, Unraveling the knot concordance group, Mem. Amer. Math. Soc. 192 (1977). Zbl0366.57005

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.