2,345 research outputs found
Dihedral coverings of trigonal curves
We classify and study trigonal curves in Hirzebruch surfaces admitting
dihedral Galois coverings. As a consequence, we obtain certain restrictions on
the fundamental group of a plane curve~ with a singular point of
multiplicity
Computing images of Galois representations attached to elliptic curves
Let E be an elliptic curve without complex multiplication (CM) over a number
field K, and let G_E(ell) be the image of the Galois representation induced by
the action of the absolute Galois group of K on the ell-torsion subgroup of E.
We present two probabilistic algorithms to simultaneously determine G_E(ell) up
to local conjugacy for all primes ell by sampling images of Frobenius elements;
one is of Las Vegas type and the other is a Monte Carlo algorithm. They
determine G_E(ell) up to one of at most two isomorphic conjugacy classes of
subgroups of GL_2(Z/ell Z) that have the same semisimplification, each of which
occurs for an elliptic curve isogenous to E. Under the GRH, their running times
are polynomial in the bit-size n of an integral Weierstrass equation for E, and
for our Monte Carlo algorithm, quasi-linear in n. We have applied our
algorithms to the non-CM elliptic curves in Cremona's tables and the
Stein--Watkins database, some 140 million curves of conductor up to 10^10,
thereby obtaining a conjecturally complete list of 63 exceptional Galois images
G_E(ell) that arise for E/Q without CM. Under this conjecture we determine a
complete list of 160 exceptional Galois images G_E(ell) the arise for non-CM
elliptic curves over quadratic fields with rational j-invariants. We also give
examples of exceptional Galois images that arise for non-CM elliptic curves
over quadratic fields only when the j-invariant is irrational.Comment: minor edits, 47 pages, to appear in Forum of Mathematics, Sigm
Lower algebraic K-theory of certain reflection groups
For a finite volume geodesic polyhedron P in hyperbolic 3-space, with the
property that all interior angles between incident faces are integral
submultiples of Pi, there is a naturally associated Coxeter group generated by
reflections in the faces. Furthermore, this Coxeter group is a lattice inside
the isometry group of hyperbolic 3-space, with fundamental domain the original
polyhedron P. In this paper, we provide a procedure for computing the lower
algebraic K-theory of the integral group ring of such Coxeter lattices in terms
of the geometry of the polyhedron P. As an ingredient in the computation, we
explicitly calculate some of the lower K-groups of the dihedral groups and the
product of dihedral groups with the cyclic group of order two.Comment: 35 pages, 2 figure
The surgery obstruction groups of the infinite dihedral group
This paper computes the quadratic Witt groups (the Wall L-groups) of the
polynomial ring Z[t] and the integral group ring of the infinite dihedral
group, with various involutions. We show that some of these groups are infinite
direct sums of cyclic groups of order 2 and 4. The techniques used are
quadratic linking forms over Z[t] and Arf invariants.Comment: Published by Geometry and Topology at
http://www.maths.warwick.ac.uk/gt/GTVol8/paper29.abs.htm
A computation of modular forms of weight one and small level
We report on a computation of holomorphic cuspidal modular forms of weight one and small level (currently level at most 1500) and classification of them according to the projective image of their attached Artin representations. The data we have gathered, such as Fourier expansions and projective images of Hecke newforms and dimensions of space of forms, is available in both Magma and Sage readable formats on a webpage created in support of this project
Tetrahedral mesh improvement using moving mesh smoothing, lazy searching flips, and RBF surface reconstruction
Given a tetrahedral mesh and objective functionals measuring the mesh quality
which take into account the shape, size, and orientation of the mesh elements,
our aim is to improve the mesh quality as much as possible. In this paper, we
combine the moving mesh smoothing, based on the integration of an ordinary
differential equation coming from a given functional, with the lazy flip
technique, a reversible edge removal algorithm to modify the mesh connectivity.
Moreover, we utilize radial basis function (RBF) surface reconstruction to
improve tetrahedral meshes with curved boundary surfaces. Numerical tests show
that the combination of these techniques into a mesh improvement framework
achieves results which are comparable and even better than the previously
reported ones.Comment: Revised and improved versio
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