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We study the numerical properties of three types of lattices constructed by means of the trace form in cyclotomic number fields. We calculate their minimum and minimal vectors, and determine whether or not they are perfect or eutactic. The lattices considered are: certain even unimodular lattices, constructed by Eva Bayer, of minimum 4 and dimension 24 (Leech lattice), 32 and 48; certain lattices related to the Leech lattice; and Craig's lattices, constructed using the successive powers of the ideal above $p$ in the $p$-th cyclotomic field.
On étudie les propriétés numériques de trois classes de réseaux construits à l'aide de la forme trace dans des corps de nombres cyclotomiques. Des algorithmes adaptés ont permis de calculer leur minimum, le nombre de vecteurs minimaux, et de déterminer s'ils sont parfaits ou eutactiques. Les réseaux considérés sont des réseaux unimodulaires pairs de minimum 4 construits par Eva Bayer en dimension 24 (Leech), 32 et 48, puis certains réseaux liés au réseau de Leech, et enfin les réseaux de Craig qui sont construits sur les puissances successives de l'idéal au-dessus de $p$ dans le $p$-ième corps cyclotomique.
We minimize a discrete version of the squared mean curvature integral for polyhedral surfaces in three-space, using Brakke's Surface Evolver [Brakke 1992]. Our experimental results support the conjecture that the smooth minimizers exist for each genus and are stereographic projections of certain minimal surfaces in the three-sphere.
Conjecturally, any "algebraic'' automorphic representation on $\GL(n)$ should have an $n$-dimensional Galois representation attached. Many examples of algebraic automorphic representations come from the cohomology over $\bold C$ of congruence subgroups of $\GL(n,\bold Z)$. On the other hand, the first author has conjectured that for any Hecke eigenclass in the mod $p$ cohomology of a congruence subgroup of $\GL(n,\Z)$ there should be an attached $n$-dimensional Galois representation.
By computer, we found Hecke eigenclasses in the mod $p$ cohomology of certain congruence subgroups of $\SL(3,\bold Z)$. In a range of examples, we then found a Galois representation (uniquely determined up to isomorphism by our data) that seemed to be attached to the Hecke eigenclass.
We describe a computer program, based on Maple, that decides whether or not a polynomial function has a simple or unimodal singularity at the origin, and determines the $\KK$-class of this singularity. The program applies the splitting lemma to the function, in an attempt to reduce the number of variables. Then, in the more interesting cases, linear coordinate changes reduce the 3-jet of the function (or the 4-jet if necessary) to a standard form, and auxiliary procedures complete the classification by looking at higher-order terms. In particular, the reduction procedure classifies cubic curves in $\P^2$.
We determine the totally real algebraic number field $F$ of degree 6 with Galois group $A_5$ and minimum discriminant, showing that it is unique up to isomorphism and that it is generated by a root of the polynomial $$ f(t) = t^6 - 10 t^4 + 7 t^3 + 15 t^2 - 14 t + 3 $$ over the rationals. We also list the fundamental units and class number of $F$, as well as data for several other fields that arose in our computations and that might be of interest.
We report on our implementation of an algorithm due to Neumann and Praeger for deciding whether or not a matrix group over a finite field contains the special linear group. This is a Monte Carlo algorithm, and thus has a small but precise probability of returning the wrong answer; this probability can be specified in advance by the user. The algorithm requires the selection of random elements from the group, and the most important problem that arose in the implementation was to find a satisfactory procedure for making this selection.