HTML and/or PDF files in the folder paplist-mt
For an html and pdf (or ppt) file with the same name, the html is an exposition. Click on any of the [ 10] items below.
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1. 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 are to the dihedral group D_p. mt-overview.html

2. 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

3. 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. fried-kop97.html %-%-% fried-kop97.pdf

4. 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. h4-0104289.html %-%-% h4-0104289.pdf

5. Alternate version of Hurwitz monodromy, spin separation, ..., for those for whom the archive version – rotated at 90 degrees – is difficult. This version is corrected up to the date listed as part of the name. [ 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

6. Thesis of Paul Bailey, 2002: Incremental Ascent of a Modular Tower via Branch Cycle Designs. Includes refined analysis of the Modular Tower defined by (A₄,± 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

7. 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=G₀ (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):

1. Whether there is a non-empty MT over a given Hurwitz space component at level 0; and
2. whether all cusps above a given level 0 o-p' cusp are p-cusps.

The diophantine discussions of §5 remind how Demjanenko-Manin worked on modular curve towers, to contrast why we still need Falting's Thm. to conclude the Main MT conjecture when the p-Frattini module has dimension exceeding 1 (G₀ is not p-super singular). rims-rev.html %-%-% rims-rev.pdf

8. 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

9. 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

10. Connectedness of families of sphere covers of A_n-Type, preprint 04/29/08. 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

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