Cyclotomic quadratic forms

@article{Sigrist2000,
abstract = {Voronoï ’s algorithm is a method for obtaining the complete list of perfect $n$-dimensional quadratic forms. Its generalization to $G$-forms has the advantage of running in a lower-dimensional space, and furnishes a finite, and complete, classification of $G$-perfect forms ($G$ is a finite subgroup of $GL (n, \mathbb \{Z\}))$. We study the standard, $\phi (m)$-dimensional irreducible representation of the cyclic group $C_m$ of order $m$, and give the, often new, densest $G$-forms. Perfect cyclotomic forms are completely classified for $\phi (m) < 16$ and for $m = 17$. As a consequence, we obtain precise upper bounds for the Hermite invariant of cyclotomic forms in this range. These bounds are often better than the known or conjectural values of the Hermite constant for the corresponding dimensions ; this is indeed the case for $m = 5,7,11,13,16,17,36$. The complete results can be taken from http://www.unine.ch/math.},
author = {Sigrist, François},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {cyclotomic quadratic forms; Voronoi's algorithm; complete classification of perfect cyclotomic forms; uper bounds; Hermite invariant of cyclotomic forms; Hermite constant},
language = {eng},
number = {2},
pages = {519-530},
publisher = {Université Bordeaux I},
title = {Cyclotomic quadratic forms},
url = {http://eudml.org/doc/248479},
volume = {12},
year = {2000},
}

TY - JOUR
AU - Sigrist, François
TI - Cyclotomic quadratic forms
JO - Journal de théorie des nombres de Bordeaux
PY - 2000
PB - Université Bordeaux I
VL - 12
IS - 2
SP - 519
EP - 530
AB - Voronoï ’s algorithm is a method for obtaining the complete list of perfect $n$-dimensional quadratic forms. Its generalization to $G$-forms has the advantage of running in a lower-dimensional space, and furnishes a finite, and complete, classification of $G$-perfect forms ($G$ is a finite subgroup of $GL (n, \mathbb {Z}))$. We study the standard, $\phi (m)$-dimensional irreducible representation of the cyclic group $C_m$ of order $m$, and give the, often new, densest $G$-forms. Perfect cyclotomic forms are completely classified for $\phi (m) < 16$ and for $m = 17$. As a consequence, we obtain precise upper bounds for the Hermite invariant of cyclotomic forms in this range. These bounds are often better than the known or conjectural values of the Hermite constant for the corresponding dimensions ; this is indeed the case for $m = 5,7,11,13,16,17,36$. The complete results can be taken from http://www.unine.ch/math.
LA - eng
KW - cyclotomic quadratic forms; Voronoi's algorithm; complete classification of perfect cyclotomic forms; uper bounds; Hermite invariant of cyclotomic forms; Hermite constant
UR - http://eudml.org/doc/248479
ER -