HTML and/or PDF files in the folder paplist-mt
paplist-mt: Generalizing modular curve properties to Modular Towers: Profinite Geometry and related Homological Algebra, the Strong Torsion Conjecture For an html and pdf (or ppt) file with the same name, the html is an exposition. Click on any of the [ 14] items below.
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1. Modular Tower Time Line: List of the papers for particular events in the development of M(odular)T(ower)s. The exposition interlaces phrases from modular curves and the Inverse Galois Problem, including reference to other files for the full story. The three major periods to date.
  1. Lessons from Dihedral groups – Before '95: Enhancing the connection between (involution) realizations of dihedral groups and modular curves.
  2. Construction and Main Conjectures on MTs – '95-'04: Generalizing the role of dihedral groups and the prime 2, to all finite groups and the primes for which they are p-perfect; connecting to the Strong Torsion Conjecture.
  3. Progress on the Main Conjectures – '05-'09: A combinatorial definition identifies MT cusps, giving a handle on properties of tower levels; Proof of the Main Conjecture for 4 branch point covers.
By aiming at generalizations of Serre's O(pen) I(mage) T(heorem) to MTs, present MT projects entwine the general theory of abelian varieties with properties of finite simple groups. Level 0 of alternating group towers, for example, often seem without resemblance to modular curves. Yet, for many towers, just a step up and the cusp structure contains a subtree – a spire – isomorphic to that for modular curves. MTTLine-domain.html

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 are to the dihedral group Dp. mt-overview.html

Above are two short expositions on MTs
Rainbow Line
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. fried-kop97.html %-%-% fried-kop97.pdf

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 A5 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. h4-0104289.html %-%-% h4-0104289.pdf

6. 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

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 (A4,± C32, 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=G0 (Thm. 2.8), and therefore of the groups Gk,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, showing 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

Rainbow Line
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. Connectedness of families of sphere covers of An-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

12. 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:
  1. Level 1 of modular curves for any odd prime p.
  2. Level 0 and 1 of the Modular Towers for the alternating groups An, n = 5 \mod 8.
In #2, at level 0 there is nothing modular curve like in the cusps. Then, level 1 sh-incidence data shows the tower cusp tree has a subtree isomorphic to that of modular curves. Then, starting at level 1 sh-incidence data shows the tower cusp tree has a subtree isomorphic to that of modular curves. The technique extends a Fried-Serre formula for lifting invariants, applied to data on cusps. Conclude: The geometric genus grows with the level, and the Main Modular Tower Conjecture holds: no rational points at high levels. CSCshInc.pdf

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