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1. CMB 2012 (vol 56 pp. 570)
Conjugacy Classes and Binary Quadratic Forms for the Hecke Groups In this paper we give a lower bound
with respect to block length
for the trace of non-elliptic conjugacy classes
of the Hecke groups.
One consequence of our bound
is that there are finitely many
conjugacy classes of a given trace in any Hecke group.
We show that another consequence of our bound
is that
class numbers are finite for
related hyperbolic \( \mathbb{Z}[\lambda] \)-binary quadratic forms.
We give canonical class representatives
and calculate class numbers
for some classes of hyperbolic \( \mathbb{Z}[\lambda] \)-binary quadratic forms.
Keywords:Hecke groups, conjugacy class, quadratic forms Categories:11F06, 11E16, 11A55 |
2. CMB 2011 (vol 54 pp. 645)
An Extension of Craig's Family of Lattices Let $p$ be a prime, and let $\zeta_p$ be a primitive $p$-th root of
unity. The lattices in Craig's family are $(p-1)$-dimensional and
are geometrical representations of the integral $\mathbb
Z[\zeta_p]$-ideals $\langle 1-\zeta_p \rangle^i$, where $i$ is a
positive integer. This lattice construction technique is a powerful
one. Indeed, in dimensions $p-1$ where $149 \leq p \leq 3001$,
Craig's lattices are the densest packings known. Motivated by this,
we construct $(p-1)(q-1)$-dimensional lattices from the integral
$\mathbb Z[\zeta _{pq}]$-ideals $\langle 1-\zeta_p \rangle^i \langle
1-\zeta_q \rangle^j$, where $p$ and $q$ are distinct primes and $i$
and $j$ are positive integers. In terms of sphere-packing density,
the new lattices and those in Craig's family have the same
asymptotic behavior. In conclusion, Craig's family is greatly
extended while preserving its sphere-packing properties.
Keywords:geometry of numbers, lattice packing, Craig's lattices, quadratic forms, cyclotomic fields Categories:11H31, 11H55, 11H50, 11R18, 11R04 |
3. CMB 1998 (vol 41 pp. 71)
Splitting patterns and trace forms The splitting pattern of a quadratic form $q$ over
a field $k$ consists of all distinct Witt indices that occur for $q$
over extension fields of $k$. In small dimensions, the complete list
of splitting patterns of quadratic forms is known. We show that
{\it all\/} splitting patterns of quadratic forms of dimension at
most nine can be realized by trace forms.
Keywords:Quadratic forms, Witt indices, generic splitting. Category:11E04 |