Cyclotomic modular lattices

Eva Bayer-Fluckiger

Journal de théorie des nombres de Bordeaux (2000)

  • Volume: 12, Issue: 2, page 273-280
  • ISSN: 1246-7405

Abstract

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Several interesting lattices can be realised as ideal lattices over cyclotomic fields : some of the root lattices, the Coxeter-Todd lattice, the Leech lattice, etc. Many of these are modular in the sense of Quebbemann. The aim of the present paper is to determine the cyclotomic fields over which there exists a modular ideal lattice. We then study an especially simple class of lattices, the ideal lattices of trace type. The paper gives a complete list of modular ideal lattices of trace type defined on cyclotomic fields.

How to cite

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Bayer-Fluckiger, Eva. "Cyclotomic modular lattices." Journal de théorie des nombres de Bordeaux 12.2 (2000): 273-280. <http://eudml.org/doc/248498>.

@article{Bayer2000,
abstract = {Several interesting lattices can be realised as ideal lattices over cyclotomic fields : some of the root lattices, the Coxeter-Todd lattice, the Leech lattice, etc. Many of these are modular in the sense of Quebbemann. The aim of the present paper is to determine the cyclotomic fields over which there exists a modular ideal lattice. We then study an especially simple class of lattices, the ideal lattices of trace type. The paper gives a complete list of modular ideal lattices of trace type defined on cyclotomic fields.},
author = {Bayer-Fluckiger, Eva},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {integral lattices; cyclotomic field; modular ideal lattice; similarity norm},
language = {eng},
number = {2},
pages = {273-280},
publisher = {Université Bordeaux I},
title = {Cyclotomic modular lattices},
url = {http://eudml.org/doc/248498},
volume = {12},
year = {2000},
}

TY - JOUR
AU - Bayer-Fluckiger, Eva
TI - Cyclotomic modular lattices
JO - Journal de théorie des nombres de Bordeaux
PY - 2000
PB - Université Bordeaux I
VL - 12
IS - 2
SP - 273
EP - 280
AB - Several interesting lattices can be realised as ideal lattices over cyclotomic fields : some of the root lattices, the Coxeter-Todd lattice, the Leech lattice, etc. Many of these are modular in the sense of Quebbemann. The aim of the present paper is to determine the cyclotomic fields over which there exists a modular ideal lattice. We then study an especially simple class of lattices, the ideal lattices of trace type. The paper gives a complete list of modular ideal lattices of trace type defined on cyclotomic fields.
LA - eng
KW - integral lattices; cyclotomic field; modular ideal lattice; similarity norm
UR - http://eudml.org/doc/248498
ER -

References

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  1. [1] C. Bachoc, C. Batut, Etude Algorithmique de Réseaux Construits avec la Forme Trace. Exp. Math.1 (1992), 184-190. Zbl0787.11024MR1203873
  2. [2] C. Batut, H.-G. Quebbemann, R. Scharlau, Computations of Cyclotomic Lattices. Exp. Math.4 (1995), 175-179. Zbl0873.11026MR1387475
  3. [3] E. Bayer-Fluckiger, Definite unimodular lattices having an automorphism of given characteristic polynomial. Comment. Math. Helv.59 (1984), 509-538. Zbl0558.10029MR780074
  4. [4] E. Bayer-Fluckiger, Lattices, cyclic group actions and number fields. In preparation. 
  5. [5] E. Bayer-Fluckiger, Lattices and number fields. Comtemp. Math.241 (1999), 69-84. Zbl0951.11016MR1718137
  6. [6] E. Bayer-Fluckiger, Ideal lattices. To appear. MR1975451
  7. [7] E. Bayer-Fluckiger, J. Martinet, Réseaux liés à des algèbres semi-simples. J. reine angew. Math.415 (1994), 51-69. Zbl0801.11020MR1277294
  8. [8] M. Craig, Extreme forms and cyclotomy. Mathematika25 (1978), 44-56. Zbl0395.10038MR491524
  9. [9] M. Craig, A cyclotomic construction of Leech's lattice. Mathematika25 (1978), 236-241. Zbl0413.10024MR533130
  10. [10] J. Martinet, Les réseaux parfaits des espaces euclidiens, Masson (1996). Zbl0869.11056MR1434803
  11. [11] H.-G. Quebbemann, Modular Lattices in Euclidean Spaces. J. Number Theory54 (1995), 190-202. Zbl0874.11038MR1354045
  12. [12] J.-P. Serre, Cours d'arithmétique. P.U.F. (1970). Zbl0225.12002

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