Aims and Topics:
The aim is to present the recent progress on the arithmetic of covers
and fundamental groups and to initialize new research in this area.
We wish to promote approaches combining moduli space ideas
with profinite Galois and group theory. A sub-program of the
conference will focus on the main theme of the ``Profinite Arithmetic
Geometry'' project , which is to investigate the connections between
the Regular Inverse Galois Problem, the Strong Torsion Conjecture
on abelian varieties over number fields and the profinite Modular
(*) Geometric Galois Theory and related structure of profinite
fundamental or Galois groups
(*) torsion on abelian varieties, l-adic representations, versions of
Serre's Open Image Theorem
(*) arithmetic and geometry of moduli spaces (e.g., Hurwitz
spaces or Shimura varieties)
(*) applications of profinite groups of finite cohomological dimension
like Demuskin and p-Poincare dual groups
(*) anabelian geometry Mathematical background:
The relation between the (regular) Inverse Galois
Problem and modular curves has grown much tighter in the last two years
using a modular curve generalization called Modular Towers (MTs). Example:
Recent results of Anna Cadoret observed that Modular Towers adds
structure to the Torsion conjecture for abelian
varieties of a given dimension. (That they collectively have few torsion
points over a number field.) In the opposite direction, the Torsion
Conjecture implies the weak Main Conjecture of MTs (no rational points at
high tower levels), a discovery tool for the Inverse Galois Problem.
A famous Shafarevich Conjecture says the cyclotomic numbers have pro-free
absolute Galois group. There are new insights into a generalizing
conjecture: An Hilbertian projective subfield F of the algebraic numbers
has pro-free absolute Galois group G_F (1992 Fried-Voelklein Annals
paper). We now see that solving this, as Fried-Voelklein does under the
stronger PAC assumption, would allow presenting G_Q usefully for many
Profinite aspects come from the profinite geometry of MTs and related
constructions. These interrelate G_K actions on their points with
profinite groups constructed canonically from all finite groups.
see also Mike Fried's page Future Scope:
We are trying to create a repository of information --
small, accessible, pieces of mathematics rather than just papers --
the web site should include the following types of data.
1. DICTIONARY OF TERMS: This would be basic definitions with examples.
For example, here are definitions one needs for talks. One wishes
these were easily available -- detached from papers -- on a web site.
Universal p-Frattini cover, Modular Towers, Harbater patching,
tangential base points and associated G_Q action along towers of moduli
spaces, GT relations, cusp types, braid rigidity, Thompson-tuples,
theta nulls ...
2. CONJECTURES and RELATIONS BETWEEN THEM: For example,
tools to understand the significance of Mazur-Merel and its
relation to the Inverse Galois Problem. For this it would be
greatly helpful to have small expositions on the following topics. Debes'
conjectured generalization of R(egular)I(nverse)G(alois)P(roblem) and
Beckman-Black, Fried-Voelklein (projective + Hilbertian implies
pro-free), Strong Torsion and Main Modular Tower Conjecture, where GT
relations show up in Modular Tower levels, ...
3. CATALOG OF PROGRESS: An archive of relevant papers defining and
achieving #1 and #2 (using convenient URLS).
see Mike Fried's page Support Acknowledgement: Special thanks to
RIMS International Research Project ``Arithmetic Algebraic
Laboratoire Painleve, Univ. Lille 1
University of California, Berkeley
JSPS Grants-in-Aid for Sci. Res. (Okayama University: Y.Yoshino, H.-F.Yamada)