Regular realizations of p-projective quotients and modular curve-like towers



Part I: Use of conjugacy classes

After explaining Nielsen classes, describes the Branch Generation Theorem. This is an addition to Conway-Fried-Parker-Voelklein from the appendix of [inv_gal.pdf]. The gist: Given conjugacy classes in a centerless group G, the regular realization of groups over a number field K using just those conjugacy classes is equivalent to one of infinitely many Hurwitz spaces defined by Nielsen classes supported just by those conjugacy classes having a K point. Further, though the Hurwitz spaces may not be irreducible, for infinitely many of the spaces, we know exactly what are their absolutely irreducible components, and their corresponding fields of definition.  An example of this result is described, the realization of alternating groups by 3-cycle branch covers, where we know precisely the components (all defined over Q) of the Hurwitz spaces of r 3-cycle covers (see [hf-can0611591.pdf, Thm. A and B]).

Part II. Is the RIGP really so hard?

[RIGP.html] has a survey of the scope of the R(egular) I(nverse) G(alois) P(roblem). The talk surveys the braid rigidity techniques that arose for covers with r conjugacy classes: with r=3 (what is called rigidity); and techniques of Thompson-Voeklein producing Hurwitz spaces with r (large) conjugacy classes, that are small covers of the configuration space, projective space Pr minus its discriminant locus. Finally, what is especially learned from the case that immediately compares with modular curves, r=4, where reduced Hurwitz spaces are upper half-plane quotients.

Part III: The RIGP realm using virtually pro-p groups

How the universal p-Frattini cover of a finite group poses a question about the RIGP that is the analog of the Fontain-Mazur conjecture. That is, if you just bound the number of branch points, then regular realization of all p-Frattini extensions of G forces the existence of a Modular Tower with this extremely unlike modular curves property. All MT levels have a K point where K is a number field. The Main Conjecture is that such K points can't exist [mt-overview.html].

Part IV. Cusps on curve components (r = 4)

There is a combinatorial description of cusps on all reduced Hurwitz spaces, and a classification of them useful for considering the Main Conjecture. A general question is the existence of p cusps. In the case of r=4 the Main Conjecture follows if you can show some tower level has at least three p cusps, or if the genus of the curve at that level exceeds 1, that there is at least one p cusp [lum-fried0611594pap.pdf, Thm. 5.1]. [h4-0104289.pdf, Chap. 9] describes in great detail the cusps of particular cases where G=A4 and A5, and where the Main Conjecture holds. In these cases the structure of the first two levels of the Modular Towers are understand in a fashion analogous to the refined understanding of modular curve levels. [rims-fried10-26-06.pdf] describes infinitely many Nielsen class cases where we now know the Main Conjecture holds.

Part V: Compare modular curve cusps with MT cusps

This is a brief review of the complete description of modular curve cusps from a MT viewpoint. A full treatment is at [London1-ModCurves.pdf]. The most important lesson: You see how modular curve properties come from p cusps generated by a Harbater-Mumford cusp. This phenomenon is known to happen in many of the Modular Towers  coming from pure-cycle Nielsen classes used by Liu and Osserman where the absolute Hurwitz spaces are connected.

Part VI. Where is the Main Conjecture with r = 4?

The description above of modular curve type cusps happens when n≡ 1 mod 5, and the Nielsen classes consist of four repetitions of the class of (n+1)/2  cycles in An. This is a consequence of using the Spin-Lift Invariant formula from the appendix as it is developed in [hf-can0611591.pdf, § 2]. In some other pure-cycle cases we can still prove the Main Conjecture. Still others do not yet yield the Main Conjecture. Cadoret has shown the Strong Torsion Conjecture implies the Main Conjecture [annaCad-Projart.pdf]. So, this starts a project of test cases for implications for the STC. For these cases we have precise results about the spaces that arise, and so it is a guide to more precise questions on the STC.
Below this line, material in the pdf file was not delivered during the actual talk.

Part VII. What happens in real MT levels!

At first it seems like it might be difficult to understand the complete set of cusp data. We have found, however, there is a concise graphical device called the sh(ift)-incidence matrix [h4-0104289.pdf, § 2.10, and examples beyond level 0 in  Chap. 9]. This discussion illustrates using the sh-incidence matrix. Especially, for example, how we see in App. B why certain MT levels are not modular curves.

Part VIII. Generalizing Serre's OIT and the g-p conjecture


App. A: Fried-Serre Formula for Spin-Lift Invariant


App. B: sh-incidence Matrix for (A4,C±32)