16,476 research outputs found
Single-Class Genera of Positive Integral Lattices
We give an enumeration of all positive definite primitive Z-lattices in
dimension >= 3 whose genus consists of a single isometry class. This is
achieved by using bounds obtained from the Smith-Minkowski-Siegel mass formula
to computationally construct the square-free determinant lattices with this
property, and then repeatedly calculating pre-images under a mapping first
introduced by G. L. Watson. We hereby complete the classification of
single-class genera in dimensions 4 and 5 and correct some mistakes in Watson's
classifications in other dimensions. A list of all single-class primitive
Z-lattices has been compiled and incorporated into the Catalogue of Lattices
On the number of perfect lattices
We show that the number of non-similar perfect -dimensional
lattices satisfies eventually the
inequalities for arbitrary
smallstrictly positive
Quaternary quadratic lattices over number fields
We relate proper isometry classes of maximal lattices in a totally definite
quaternary quadratic space (V,q) with trivial discriminant to certain
equivalence classes of ideals in the quaternion algebra representing the
Clifford invariant of (V,q). This yields a good algorithm to enumerate a system
of representatives of proper isometry classes of lattices in genera of maximal
lattices in (V,q)
Characteristic Lie rings, finitely-generated modules and integrability conditions for 2+1 dimensional lattices
Characteristic Lie rings for Toda type 2+1 dimensional lattices are defined.
Some properties of these rings are studied. Infinite sequence of special kind
modules are introduced. It is proved that for known integrable lattices these
modules are finitely generated. Classification algorithm based on this
observation is briefly discussed.Comment: 11 page
Finite quotients of Z[C_n]-lattices and Tamagawa numbers of semistable abelian varieties
We investigate the behaviour of Tamagawa numbers of semistable principally
polarised abelian varieties in extensions of local fields. In view of the
Raynaud parametrisation, this translates into a purely algebraic problem
concerning the number of -invariant points on a quotient of -lattices
for varying subgroups of and integers . In
particular, we give a simple formula for the change of Tamagawa numbers in
totally ramified extensions (corresponding to varying ) and one that
computes Tamagawa numbers up to rational squares in general extensions.
As an application, we extend some of the existing results on the -parity
conjecture for Selmer groups of abelian varieties by allowing more general
local behaviour. We also give a complete classification of the behaviour of
Tamagawa numbers for semistable 2-dimensional principally polarised abelian
varieties, that is similar to the well-known one for elliptic curves. The
appendix explains how to use this classification for Jacobians of genus 2
hyperelliptic curves given by equations of the form , under some
simplifying hypotheses.Comment: Two new lemmas are added. The first describes permutation
representations, and the second describes the dependence of the B-group on
the maximal fixpoint-free invariant sublattice. Contact details and
bibliographic details have been update
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