lum03-12-04.pdf and related files: Slides for talk at Luminy March 12, 2004

In addition to the Luminy talk you may select other files related to that talk. Click on abstract for a short description.

Profinite geometry: Higher rank Modular Towers
16 slides from the upcoming talk in Luminy. Other related talks:

Matthieu Romagny and Stefan Wewers will introduce Nielsen classes and material on Hurwitz spaces.
Helmut Voelklein and Kay Magaard will introduce Hurwitz monodromy action, necessary for computations.
Pierre Debes will give the definition of a (rank 0) Modular Tower, comparing that with modular curves.
The (weak; rank 0) Main Conjecture is that there are no rational points at suitably high levels. Pierre will reduce this conjecture, for four branch point towers, to showing the genus rises with the levels.
Darren Semmen will present the profinite Frattini category. In this framework he will show how Schur multipliers control properties of the Modular Tower levels.

Luminy abstract
lum03-12-04.pdf file; or return to the home page.

Modular Towers: (Pro-)finite Groups and Cusp Geometry: 25 slides from the Talk in Durham, Noncommutative aspects of Number Theory, Aug. 28--Sept. 5, 2003, last revision: 10/22/2003. Durham abstract
durhamsh09-30-03.pdf file.

Two genus 0 problems of John Thompson: To appear in the Cambridge University Press volume dedicated to John Thompson's 70th birthday, Conference November, 2002, last revision: 02/10/2004. Thompson, genus 0 abstract
thomp-genus0.pdf file.

Extension of constants series and towers of exceptional covers: Paper from the cryptology conference at University of Florida, March 2003, Last revision: 02/20/2004. Extension of constants abstract: This is the abstract I am using for talks at Lille and Bordeaux after the conference.
exceptTow.pdf file.

Schur multiplier types and Shimura-like systems of varieties: Paper with Darren Semmen containing the full results on Modular Towers related to the Durham and Luminy talks, Last revision: 02/20/2004. Shimura-like systems abstract
schurtype.pdf file.

Hurwitz monodromy, spin separation and higher levels of a Modular Tower: Paper with Paul Bailey, background on all tools necessary for Modular Towers. Includes proof of the Main Conjecture for the A₅, p=2 Modular Tower, and related towers. Appeared on pages 79--221 of MSRI volume (below). Last revision: 04/24/2002. Spin separation and higher levels abstract
h4.pdf file.

Arithmetic Fundamental Groups and Noncommutative Algebra: Prelude to the volume produced for the MSRI conference Fall 1999 of the same name. Appeared as Proceedings of Symposia in Pure Mathematics, 70 (2002) editors M. Fried and Y. Ihara, 1999 von Neumann Conference on Arithmetic Fundamental Groups and Noncommutative Algebra, August 16-27, 1999 MSRI, vii--xxx. Last revision: 04/24/2002. MSRI Volume abstract
msrivol01.pdf file.

For getting Ghoscript and Adobe Reader versions, along with comments on their use, click here.

Abstracts for pdf files

Abstract: Outline proof of Modular Towers' Main Conjecture and its higher rank generalization

The genus g of a curve discretely separates decidedly different two variable algebraic relations. So, we can focus on the connected moduli space Mg. Yet, modern applications require a data variable (function) on the curve. The resulting spaces are versions, depending on what we need from this data variable, of Hurwitz spaces. A Nielsen class consists of r ≥ 3 conjugacy classes C in the data variable monodromy G. It gives a striking genus analog.

Using Frattini covers of G, every Nielsen class produces a projective system of reduced spaces (and corresponding Nielsen classes) for any prime p dividing |G|. This is the Modular Tower (MT) attached to (G,C,p), and its components form a tree of of irreducible (dim r-3) components. Some branches may be finite, but the (weak) Main Conjecture says, if G is p-perfect}, there are no rational points at high tower levels (of an infinite branch).

Modular curve towers (the simplest case; G is dihedral) are an avatar with just one (infinite) branch where the conjecture has long been known. The case
r=4 is closest to classical because the MT s (minus their cusps) are systems of upper half plane quotients covering the j-line. These are our topics.

• Identifying maximal projective systems of components on a MT\ and labeling them through g-p', Weigel and p cusp branches.
• Outline proof of the (weak) Main Conjecture when r=4.
• Definitions, examples and a (strong) Main Conjecture for higher rank MT s: almost all primes produce a modular curve-like system.

Abstract: Higher rank Modular Towers:

Frattini extensions of a finite group G are the objects behind Modular Towers. Let p be a prime dividing |G| and C a collection of four p' conjugacy classes of G. Pierre Debes' talk defined from this a projective sequence H_k, k=0,1,... of curves. Each reduced Hurwitz space H_k is an upper half-plane quotient and j-line cover. Pierre reduced the weak conjecture -- no rational points at high levels -- on Modular Towers to this: For large k, all components of H_k have genus exceeding 1.

You must know three things to compute the genus of these components.

What are the H_k components.
What are the widths of cusps (points over infinity) in each component.
What are the ramified points in each component over the elliptic points (j=0 or 1).

Reduced Nielsen classes let us calculate components, and cusp and elliptic ramification. We'll see how the Frattini property controls growth of cusp widths with k, outlining a proof of the (weak; rank 0) conjecture.

Modular Tower components are moduli spaces, though mostly not modular curves. So congruence subgroups of PSL₂(Z) don't define them. Their virtues are in the useful moduli problems they define, and how their applications benefit from modular curve thinking. That direction uses higher rank Modular Towers where there is a strong Conjecture . This generalizes that among all modular curves only finitely many have genus 0 or 1. We show how this works with two contrasting examples.

Serre's open image theorem interprets as a statement about the easiest rank 2 Modular Tower. This is from Z/2 acting on F₂ (a free group on two generators). We call this the Z/2 case. The other case is for a Modular Tower from Z/3 acting on F₂ (the Z/3 case). Certain cusps give special degeneration for the moduli of their components. We call these, g(roup)-p' cusps. Example: Shifts of H(arbater)-M(umford) cusps; both the Z/2 and Z/3 cases have these.

Projective sequences of g-p' components should have an analytic expression in k for their genus. This is a strong statement. Underlying it is conjectures relating components to the cusps they contain. If true, the whole apparatus of objects like Tate curves and tangential base points will apply to general Modular Towers.

Components of genus 0 or 1 are especially useful, particularly to the Inverse Galois Problem. We conclude with properties of the Z/3 case for the prime 2.

A Theorem of Andr'e says, unlike the Z/2 case, only finitely many points are (in the Shimura sense) special.
Different arithmetic behavior on distinct components appears for levels k=0,1.
Two genus 1 components for level 1 give the last chance for infinitely many four branch-point regular realizations of the maximal nonsplit extensions of the group A₅ by an elementary 2-group.

Abstract:Profinite groups and cusp geometry at Durham:

Introduced the generalization of H(arbater)-Mumford representatives and the organization of Modular Tower components by a general cusp type. Frattini extensions of groups are the defining property behind Modular Towers. This talk emphasizes how that Frattini property controls Modular Tower components, especially how one computes and interprets such facts as cusp widths. Suppose X_k is an upper half-plane quotient j-line cover. Then, you must know three things to compute its genus.

What are its components.
What are the widths of cusps (points over infinity) in each component.
What are the ramified points in each component over the elliptic points (j=0 or 1).

We showed how to use Frattini properties and Schur multipliers to understand each item on this list. Modular Towers uses a natural pairing on cusps, the shift-incidence matrix. This efficient gadget succinctly summarizes immense information about these non-modular curve spaces with modular curve-like properties.

Abstract: Thompson genus 0 problems:

The cases called Z/2 and Z/3 in the Luminy abstract appear here in their most elemental form. This is the easiest place to pick up the apparatus of Nielsen classes and the properties of spaces defined by them. Two well-known problems on families of polynomials illustrate all the techniques and the work of many people whose contributions are discussed in the MSRI volume.

Abstract: Extension of constants:
Let F_q be the finite field of order q. We use a simple characterization of exceptional covers (of projective nonsingular curves) over F_q: A cover maps one-one on F_q^t points for infinitely many t. Lenstra suggested there may be an Exceptional (cover) Tower. We construct and use it to organize progress and unsolved problems on genus 0 exceptional covers.

To analyze exceptional covers, you use the first term of the "extension of constants" series attached to a cover of curves. We explain that. Then we relate exceptional covers in all characteristics through the Guralnick-Thompson genus 0 problem and Serre's open image theorem. The talk will end by interpreting exceptional correspondences and Davenport pairs with zeta functions. As Davenport desired in the early '70's, this deepens ties between exceptional covers (and their related cryptology) the Weil conjectures.

Abstract: Spin separation and higher Modular Tower levels:

Most of the basics on Hurwitz spaces for use in Modular Towers appear in this paper. This includes the shift-incidence matrix, the formula for the genus of components computed from Nielsen class data, and the generalization of Serre's formula for the effect of Spin covers of finite groups on lifting invariants. We illustrated everything by showing that Main Conjecture on Modular Towers holds for a set of specific Modular Towers that had considerable motivation from the literature. Example: The A₅ Modular Tower when p=2 and the conjugacy classes for the tower were four repetitions of the 3-cycle conjugacy class. These examples appeared in a spirit akin to what one might see on a paper about specific modular curves. In this A₅ case we saw an explanation for these phenomena.

Why there was just one component at level 0 and it had genus 0.
Why there were two components at level 1, of respective genuses 12 and 9, and why all the real points fell as one connected set on the genus 12 component.
Following a formula of Serre, why the genus 12 component supports a nontrivial theta-null, and the genus 9 component does not.

Abstract: Schur multiplier types and Shimura-like systems of varieties:

Abstract: Introduction to all the papers in the MSRI volume: