Geometry
and Arithmetic of Moduli Spaces of Coverings
9-20 June 2008, Istanbul
The web page of the summer school is http://math.gsu.edu.tr/GAMSC/index.htm.
For those interested in attending, contact me or gamsc.school@gmail.com.
The lecturers of the school:
K. Aker (Istanbul)
J. Bertin (Grenoble)
M. Boggi (Jussieu)
A. Cadoret (Bordeaux)
P. Debes (Lille)
M. Emsalem (Lille)
M. Fried (Irvine)
M. Korkmaz (Ankara)
R. Litcanu (Iasi)
P. Lochak (Jussieu)
M. Romagny (Paris)
L. Schneps (Jussieu)
S. Unver (Istanbul)
S. Wewers (Hannover)
A. The Program: The
program
assumes a solid knowledge of basic algebraic geometry and topological
fundamental groups. A knowledge of moduli spaces of Riemann surfaces,
mapping class groups, and profinite groups will be helpful, but not
necessary.
1st Part: Established Background
I. Stacks and fundamental group
I.1. orbifolds, stacks and groupoids and dictionaries between them,
gerbes
I.2. algebraic fundamental group of schemes and stacks
I.3. algebraic patching
I.4. further aspects of the fundamental group
II. Geometry and arithmetic of
moduli spaces of coverings
II.1. models of curves
II.2. geometry of Hurwitz spaces (including construction)
II.3. topological, group and combinatorial aspects
II.4. reduction modulo p and compactification
II.5. connected Hurwitz space components and inverse Galois theory
2nd Part: Applications and
developments
III. Connected Hurwitz space
components and inverse Galois theory
III.1. the action of the arithmetic Galois group on the geometric
fundamental group of the thrice punctured projective line
III.2. profinite braid groups and the genus 0 GT group.
III.3. McLane relations, discrete complexes of curves and how to
pass to strictly positive genus.
III.4 profinite complexes of curves and another geometric view of
the GT group.
III.5. the contractibility conjecture and its consequences.
IV. Modular Towers (MT)
IV.1 construction and the main conjectures
IV.2. cusp and component structure on MTs
IV.3. modular curve towers as MTs, and alternating group towers
IV.4. Inverse Galois Theory, Abelian Varieties and the MT program
IV.5. MTs with a spire and generalizing Serre's Open Image Theorem
B. Relation of my talks to
Hurwitz space arithmetic and MTs:
Ist Part: The overriding
theme has
three parts:
- Why we care about identifying components of moduli
spaces (like Hurwitz spaces).
- Tools we use to identify components.
- Significant cases when we are successful at identifying
components.
Hurwitz spaces – spaces of sphere covers – start with
giving a finite group. We make serious us these
spaces, as with finite groups, starting with a
serious collection of finite groups. Our archetypal example will be modular
curves, which are Hurwitz spaces based on dihedral groups.
We understand the geometry of modular curves through their cusps
(points on the boundary of the curve). We produce functions and
forms on modular
curves through their expansions about cusps.
Advanced topics, like Serre's Open Image Theorem, and the
uniformization of elliptic curves by modular curves, made headway
by gathering new moduli information at cusps.
So, you won't be surprised that the three theme parts use cusps on
Hurwitz spaces to make their points.
For this analogy I need another class of groups. I want the group
theory – though harder – not too much
harder than that for dihedral groups. Most important: The applications
must be as down-to-earth as, while interestingly different
from, those of modular curves. Using papers and
results of J.P. Serre, I have chosen the collection of
alternating groups.
2nd Part: My Lectures:
Here is a list of the three lectures I will give. They have attached
html files. These include abstracts, as they are available, and
expositions on basic definitions and explanations of the results in
related papers.
- Dihedral
Groups: MT view of
Modular curve cusps
- Updating
an Abel-Gauss-Riemann Program: Cusp and component structure on MTs
(including the introduction of spires)
- Conway-Fried-Parker-Voelklein
connectedness results: CFPV-Thm.html
and GQpresentation.html
3rd Part: Related Lectures of
Others:
Pierre Dèbes:
Pierre's themes are the basic definitions that go into constructing
Modular Towers (MTs) and relating them to the R(egular)
I(nverse) G(alois) P(roblem). As above,
I've attached URLs of some definition files.
- p-universal
Frattini cover, definition of MT, statement of the main conjectures,
the dihedral group example: fried-kop97.html
- various forms of the conjectures and their connections:
Fried-Kopeliovich, stacks vs moduli spaces: mt-overview.html
- connection with the torsion on abelian varieties, results
and conjectures from my paper with Anna Cadoret: RIGP.html