2. Outline of how M(odular) T(ower)s has, to date, been shown to generalize modular curve towers. The basic analogy: For any prime p, all p-perfect groups (having no Z/p quotient) are to MTs for the prime p as modular curve towers for the prime p (not 2) are to the dihedral group D_{p}. mt-overview.html

Above are two short expositions on MTs Next are my papers from the first period of MT development

3. Introduction to Modular Towers: Generalizing dihedral group–modular curve connections, Recent Developments in the Inverse Galois Problem, Cont. Math., proceedings of AMS-NSF Summer Conference 1994, Seattle 186 (1995), 111–171. This version has corrections from the printed version up to about 1999. As, however, it was originally in amstex, I haven't retexed it in several years. modtowbeg.html %-%-% modtowbeg.pdf

4. with Yaacov Kopeliovic, Applying Modular Towers to the Inverse Galois Problem, Geometric Galois Actions II Dessins d'Enfants, Mapping Class Groups and Moduli 243, London Mathematical Society Lecture Note series, (1997) 172–197. We introduce two major uses of Modular Towers:

5. with Paul Bailey, Hurwitz monodromy, spin separation and higher levels of a Modular Tower, Arithmetic fundamental groups and noncommutative algebra, PSPUM vol. 70 of AMS (2002), 79–220. arXiv:math.NT/0104289 v2 16 Jun 2005. Computes everything I thought would be of interest about level one (versus level 0) of the M(odular) T(ower) attached to A_{5} and four repetitions of the conjugacy class of 3-cycles, in particular showing the Main MT Conjecture for it: No K points at high levels (K any number field). Level 0 has one component of genus 0, while level one has two components, one of genus 12, the other of genus 9. The paper includes a complete conceptual accounting of the nature of all cusps, and all real points on both components. Also, why a version of the spin cover (based on serre-oddraminv.pdf) obstructs anything beyond level 1 for the genus 9 component. Its intent: A small book archetype for knowing as much about one MT as one might know about any modular curve tower. Many of its many topics have been elaborated in the book "Monodromy, l-adic representations and the regular inverse Galois problem." Though this is not yet complete, a good portion of it has been put on-line here as of 04/04/18. Also, the freshest copy of this paper is here as of 04/04/18. h4-0104289.html %-%-% h4-0104289.pdf

6. Alternate version of Hurwitz monodromy, spin separation, ..., with further small corrections beyond the archive version, with those corrections listed in ./paplist-mt/h4-03-28-06-cor.html. [ lum-fried0611594pap.pdf, App. C] has corrected typos up to 03/28/06, and the arkiv version is close to that. h4-03-28-06.pdf

7. Thesis of Paul Bailey, 2002: Incremental Ascent of a Modular Tower via Branch Cycle Designs. Includes refined analysis of the Modular Tower defined by (A_{4},± C_{32}, p=2). While the notation of the thesis is Paul's own, we have it here to take advantage of certain remarkable examples found by Paul. When we refer to these we translate into the notation of later MT papers. pBaileyThesis2002.html %-%-% pBaileyThesis2002.pdf

8. Moduli of relatively nilpotent extensions, Inst. of Math. Science Analysis 1267, June 2002, Communications in Arithmetic Fundamental Groups, 70–94. Developed from three lectures I gave at RIMS, Spring 2001. Gives the most precise available description of the p-Frattini module for any p-perfect finite group G=G_{0} (Thm. 2.8), and therefore of the groups G_{k,ab}, k ≥ 0, from which we form the abelianized M(odular) T(ower). §4 includes a classification of Schur multiplier quotients, from which we figure two points (see the html file):

Next are papers in the period proving cases of the Main Conjectures

9. The Main Conjecture of Modular Towers and its higher rank generalization, in Groupes de Galois arithmetiques et differentiels (Luminy 2004; eds. D. Bertrand and P. Debes), Sem. et Congres, Vol. 13 (2006), 165–233. lum-fried0611594pap.html %-%-% lum-fried0611594pap.pdf

10. Regular realizations of p-projective quotients and modular curve-like towers, Oberwolfach report #25, on the conference on pro-p groups, April (2006), 64–67. Also available at the conference archive. %-%-% oberwolf-friedrep06-16-06.pdf

11. Frattini towers and the shift-incidence cusp pairing: The M(odular) T(ower) view of modular curve towers: Preprint as of 4/02/10. In parallel, we treat two cases of the ideas from MTs:

12. Connectedness of families of sphere covers of A_{n}-Type, Last Revision 7/01/14. Restricting to covers of the sphere by a compact Riemann surface of a given type, do all such compose one connected family? Or failing that, do they fall into easily discernible components? The answer has often been "Yes!," though sometimes the reason is that the families were (or close to) simple-branched covers. This paper shows the importance of the problem, and the need to adjust to a more complicated answer when the covers are not simple-branched. By using solutions of the problem, the paper gives infinitely many cases for which the Modular Tower over these spaces for the prime 2 satisfies a modular curve conclusion, the M(ain) C(onjecture): For any number field K, high tower levels have no rational points. In one way the prime 2 is the hardest case. That is, these examples – first appearing in a work of Liu and Osserman – are spaces of covers with alternating monodromy group, and the Schur multiplier of the group enters. The Fried-Serre formula shows the exact nature of the cusps on these spaces. That gives the final step in the conclusion. The paper concludes by outlining the MC for the other easier primes, doing special cases. The complete solution for those primes requires a new piece of modular representation theory. twoorbit.html %-%-% twoorbit.pdf

13. Combinatorics of Sphere Covers and the Shift-Incidence Matrix (pairing on Hurwitz space cusps), 10/08/08: A proposal to the AMS, convenient to see a quick discussion of the significance displayed in the Sh-Incidence Matrix. §4 has four examples. Two are infinite series of examples:

14. Monodromy, l-adic representations and the Regular Inverse Galois Problem (latest version 02/01/19): This is a book connecting to, and extending, key topics in two of J.P. Serre's books:

15. with Mark van Hoeij, The small Heisenberg group and l-adic representations from Hurwitz spaces, Latest version: 08/14/14. The Hurwitz space approach to the regular Inverse Galois Problem produced new serious of simple group realizations as Galois over Q, and it identified obstructions to regular realizations of covering groups of simple groups as generalizing renown results on modular curves. This was the motivation for the M(odular) T(ower) program. We use that program to explicitly construct towers, and analyze their cusps, to produce families of l-adic representations. This models properties Serre used in his O(pen) I(mage) T(heorem) on l-adic representations from projective systems of points on modular curves. Our main example regards modular curves as MTs constructed from the semidirect product of Z/2 acting Z^{2}. It then extend those techniques to Z/3 acting on Z^{2}. The identification of tower components running over all primes l runs into new components on the tower levels. Those corresponding to one prime power (level k = 0) have this qualitative description.

16. Introduction to ''Monodromy, l-adic representations and the Regular Inverse Galois Problem,'' In "Teichmuller theory and its impact,'' in the Nankai Series in Pure, Applied Mathematics and Theoretical Physics, the World Scientific Company (2018). §1 introduces Nielsen classes attached to (G,C), where C is r conjugacy classes in a finite group G, and a braid action on them. These give reduced Hurwitz spaces, denoted H(G,C)^{rd}. The section concludes with a braid formula for the genus of these spaces when r = 4. If there is at least one prime l for which G is divisible by l, but has no Z/l quotient, then there is a canonical tower of reduced Hurwitz spaces over H(G,C)^{rd}, using the Universal Frattini cover, ^{~}G, of G, and ^{~}G_{ab}, its abelianized version. The towers are nonempty assuming C are l' classes satisfying a cohomological condition from a lift invariant. A M(odular)T(ower) is a projective sequence of components of the canonical tower. §2 introduces the book [Fr18], which takes on generalizing Serre's O(pen)I(mage)T(heorem}, interpreted as the case when G is a dihedral group D_{l} and C is four repetitions of the involution conjugacy class. Serre's Theorem separated decomposition groups of projective sequences of points in the modular curve towers into two types: CM (complex multiplication) and GL_{2}. When r = 4, all MT levels are upper half-plane quotients ramified at 0 (of order 3), 1 (of order 2) and ∞ (corresponding to the cusps). They are appropriate therefore to compare Serre's OIT with the cusps and decomposition group fibers. [Fr18] emphasizes new phenomena in cusps, and components, while still showing in high tower levels a valid comparison with modular curves. It also aims to show how MTs can expand the applications usual for modular curve towers, recognizing those problems directly interpret from the Inverse Galois Problem. The l-adic representations of the title come from the abelianized version of the Universal Frattini cover. ChernPapOIT01-01-17.pdf

17. l-adicReps-RIGP: Talks notes for Oberwolfach talk ''l-adic Representations and the Regular Inverse Galois Problem,''given on 04/16/18:

18. The sh-incidence matrix and Hurwitz space orbits (preprint 12/20/18): sh-incidence.pdf