6,372 research outputs found
Algebraic curves with many automorphisms
Let be a (projective, geometrically irreducible, nonsingular) algebraic
curve of genus defined over an algebraically closed field of odd
characteristic . Let be the group of all automorphisms of which
fix element-wise. It is known that if then the -rank
(equivalently, the Hasse-Witt invariant) of is zero. This raises the
problem of determining the (minimum-value) function such that whenever
then has zero -rank. For {\em{even}} we prove
that . The {\em{odd}} genus case appears to be much more
difficult although, for any genus , if has a solvable
subgroup such that then has zero -rank and fixes a
point of . Our proofs use the Hurwitz genus formula and the Deuring
Shafarevich formula together with a few deep results from finite group theory
characterizing finite simple groups whose Sylow -subgroups have a cyclic
subgroup of index . We also point out some connections with the Abhyankar
conjecture and the Katz-Gabber covers
Singer quadrangles
[no abstract available
Representation theory for high-rate multiple-antenna code design
Multiple antennas can greatly increase the data rate and reliability of a wireless communication link in a fading environment, but the practical success of using multiple antennas depends crucially on our ability to design high-rate space-time constellations with low encoding and decoding complexity. It has been shown that full transmitter diversity, where the constellation is a set of unitary matrices whose differences have nonzero determinant, is a desirable property for good performance. We use the powerful theory of fixed-point-free groups and their representations to design high-rate constellations with full diversity. Furthermore, we thereby classify all full-diversity constellations that form a group, for all rates and numbers of transmitter antennas. The group structure makes the constellations especially suitable for differential modulation and low-complexity decoding algorithms. The classification also reveals that the number of different group structures with full diversity is very limited when the number of transmitter antennas is large and odd. We, therefore, also consider extensions of the constellation designs to nongroups. We conclude by showing that many of our designed constellations perform excellently on both simulated and real wireless channels
Finite groups acting on 3-manifolds and cyclic branched coverings of knots
We are interested in finite groups acting orientation-preservingly on
3-manifolds (arbitrary actions, ie not necessarily free actions). In particular
we consider finite groups which contain an involution with nonempty connected
fixed point set. This condition is satisfied by the isometry group of any
hyperbolic cyclic branched covering of a strongly invertible knot as well as by
the isometry group of any hyperbolic 2-fold branched covering of a knot in the
3-sphere. In the paper we give a characterization of nonsolvable groups of this
type. Then we consider some possible applications to the study of cyclic
branched coverings of knots and of hyperelliptic diffeomorphisms of
3-manifolds. In particular we analyze the basic case of two distinct knots with
the same cyclic branched covering.Comment: This is the version published by Geometry & Topology Monographs on 29
April 200
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