11 research outputs found
A GAP package for braid orbit computation, and applications
Let G be a finite group. By Riemann's Existence Theorem, braid orbits of
generating systems of G with product 1 correspond to irreducible families of
covers of the Riemann sphere with monodromy group G. Thus many problems on
algebraic curves require the computation of braid orbits. In this paper we
describe an implementation of this computation. We discuss several
applications, including the classification of irreducible families of
indecomposable rational functions with exceptional monodromy group
Computation of highly ramified coverings
An almost Belyi covering is an algebraic covering of the projective line,
such that all ramified points except one simple ramified point lie above a set
of 3 points of the projective line. In general, there are 1-dimensional
families of these coverings with a fixed ramification pattern. (That is,
Hurwitz spaces for these coverings are curves.) In this paper, three almost
Belyi coverings of degrees 11, 12, and 20 are explicitly constructed. We
demonstrate how these coverings can be used for computation of several
algebraic solutions of the sixth Painleve equation.Comment: 26 page
Hurwitz Monodromy and Full Number Fields
We give conditions for the monodromy group of a Hurwitz space over the
configuration space of branch points to be the full alternating or symmetric
group on the degree. Specializing the resulting coverings suggests the
existence of many number fields with full Galois group and surprisingly little
ramification --- for example, the existence of infinitely many such number
fields unramified away from {2,3,5}
Generating sets of Affine groups of low genus
We describe a new algorithm for computing braid orbits on Nielsen classes. As
an application we classify all families of affine genus zero systems; that is
all families of coverings of the Riemann sphere by itself such that the
monodromy group is a primitive affine permutation group
Alternating groups and moduli space lifting Invariants
Main Theorem: Spaces of r-branch point 3-cycle covers, degree n or Galois of
degree n!/2 have one (resp. two) component(s) if r=n-1 (resp. r\ge n). Improves
Fried-Serre on deciding when sphere covers with odd-order branching lift to
unramified Spin covers. We produce Hurwitz-Torelli automorphic functions on
Hurwitz spaces, and draw Inverse Galois conclusions. Example: Absolute spaces
of 3-cycle covers with +1 (resp. -1) lift invariant carry canonical even (resp.
odd) theta functions when r is even (resp. odd). For inner spaces the result is
independent of r. Another use appears in,
http://www.math.uci.edu/~mfried/paplist-mt/twoorbit.html, "Connectedness of
families of sphere covers of A_n-Type." This shows the M(odular) T(ower)s for
the prime p=2 lying over Hurwitz spaces first studied by,
http://www.math.uci.edu/~mfried/othlist-cov/hurwitzLiu-Oss.pdf, Liu and
Osserman have 2-cusps. That is sufficient to establish the Main Conjecture: (*)
High tower levels are general-type varieties and have no rational points.For
infinitely many of those MTs, the tree of cusps contains a subtree -- a spire
-- isomorphic to the tree of cusps on a modular curve tower. This makes
plausible a version of Serre's O(pen) I(mage) T(heorem) on such MTs.
Establishing these modular curve-like properties opens, to MTs, modular
curve-like thinking where modular curves have never gone before. A fuller html
description of this paper is at
http://www.math.uci.edu/~mfried/paplist-cov/hf-can0611591.html .Comment: To appear in the Israel Journal as of 1/5/09; v4 is corrected from
proof sheets, but does include some proof simplification in \S
A GAP Package for braid orbit computation, and applications
Abstract: Let G be a finite group. By Riemann’s Existence Theorem, braid orbits of generating systems of G with product 1 correspond to irreducible families of covers of the Riemann sphere with monodromy group G. Thus many problems on algebraic curves require the computation of braid orbits. In this paper we describe an implementation of this computation. We discuss several applications, including the classification of irreducible families of indecomposable rational functions with exceptional monodromy group.