Arithmetic Galois Theory and Related Moduli Spaces

This conference is located :

Date : October 23-27 & 30-31, 2006
Place: RIMS, Kyoto University, Japan
Room: RIMS Building 420 (Oct.23-27), 115 (Oct.30-31)

Program Organizers
Pierre Debes, Mike Fried, Jochen Koenigsmann, Hiroaki Nakamura, Ken Ribet

Program ( dvi / ps / pdf ) -- revised, October 19

((( list of talk titles and available abstracts )))

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 Tower Program.
(*) 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 practical problems.
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