122 research outputs found
On the simplest sextic fields and related Thue equations
We consider the parametric family of sextic Thue equations where
is an integer and is a divisor of . We
show that the only solutions to the equations are the trivial ones with
.Comment: 12 pages, 2 table
Galois theory of iterated endomorphisms
Given an abelian algebraic group over a global field , , and a prime , the set of all preimages of under some
iterate of generates an extension of that contains all
-power torsion points as well as a Kummer-type extension. We analyze the
Galois group of this extension, and for several classes of we give a simple
characterization of when the Galois group is as large as possible up to
constraints imposed by the endomorphism ring or the Weil pairing. This Galois
group encodes information about the density of primes \p in the ring of
integers of such that the order of (\alpha \bmod{\p}) is prime to .
We compute this density in the general case for several classes of ,
including elliptic curves and one-dimensional tori. For example, if is a
number field, is an elliptic curve with surjective 2-adic representation
and with , then the density of
with (\alpha \bmod{\p}) having odd order is 11/21.Comment: 33 pages; The appendix has been updated, several examples have been
redone, and a number of typos corrected. The paper has been accepted for
publication in Proceedings of the London Mathematical Societ
Lattices and automorphisms of compact complex manifolds
This work makes use of well-known integral lattices to construct complex algebraic varieties reflecting properties of the lattices. In particular the automorphism groups of the lattices are closely related to the symmetries of varieties.
The constructions are to two types: generalised Kummer manifolds and toric varieties. In both cases the examples are of the most interest.
A generalised Kummer manifold is the resolution of the quotient of a complex torus by some finite group G. A description of the construction for certain cyclic groups G by given in terms of holomorphic surgery of disc bundles. The action of the automorphism groups is given explicitly. The most important example is a compact complex 12-dimensinoal manifold associated to the Leech lattice admitting an action of the finite simple Suzuki group. All these generalised Kummer manifolds are shown to be simply connected.
Toric varieties are associated to certain decompositions of Rn into convex cones. The automorphism groups of those associated to Weyl group decompositions of Rn are calculated. These are used to construct 24-dimensional singular varieties from some Neimeier lattices. Their symmetries are extensions of Mathieu groups and their singularities closely related to the Golay codes
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