Click on any of the [ 19] items below.

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 (not 2) 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. We introduce two major uses of Modular Towers:

  1. They encode a relation between the S(trong) T(orsion) C(onjecture) on abelian varieties and the R(egular) I(nverse) G(alois) P(roblem).
  2. They produce towers of spaces based on finite groups, rather than semi-simple groups – as with Shimura varieties – whose properties generalize those of modular curves.
Item #1 is based on a geometric analog of the Fontaine-Mazur conjecture. Its truth forces the existence of Modular Towers and the Main MT Conjecture: That high tower levels have no K points over a fixed number field K. The html file exposits on how this attaches Modular Towers to each p-perfect finite group. A Timeline reviews many contributions, especially to the Main Conjecture, which has been shown as of 2009 in many cases with two quite different methods. 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. 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 (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. 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:

  1. modular curves, which here derive from the semi-direct product of Z/2 acting through multiplication by -1 on Z; and
  2. an equally rich case from Z/3 acting irreducibly on Z2.
Modular curves are families of sphere covers attached to dihedral groups. By working out general ideas from MTs in this case, we see something familier, their cusps and the monodromy on homology in a fiber, in a new way. Then, following analogous methods, a case simple enough that the group theory is no problem, but you can see how the tools extend. To take Serre's Open Image Theorem beyond modular curves, to general moduli of abelian varieties, has lacked a way to master the limiting effect of correspondences – read motives – of arithmetic monodromy on special tower fibers. Our Z/3 case shows how Frattini data in our Hurwitz space approach helps tame that structure. This paper is a prelude to twoorbit.html which gives a general approach to the main MT conjecture through the sh-incidence pairing on cusps. sh-incModCurve.pdf sh-incModCurve.pdf

12. Connectedness of families of sphere covers of An-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:

  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

14. Monodromy, l-adic representations and the Regular Inverse Galois Problem (latest version 11/01/19): This is a book connecting to, and extending, key topics in two of J.P. Serre's books:

It illustrates a general relation between l-adic representations, as in generalizing Serre's O(pen) I(mage) T(heorem), and the Regular version of the Inverse Galois Problem. These come together over the M(odular) T(ower) generalization of modular curves.

Chap. 4 explains MTs (started in 1995) as a program motivated by such a relationship. Chap. 1 and Chap. 2 include exposition tieing up threads of work coming just before MTs. They show how Hurwitz spaces both encode versions of the Regular Inverse Galois Problem and geometrically connect to classical arithmetic geometry problems. Especially, why the Inverse Galois Problem has been so difficult. We modernize investigating definition fields of components of Hurwitz spaces vis-a-vis lift invariants with examples.

Then, we interpret the OIT in a generality not indicated by Serre's approach. Chap. 5 uses one example, in the sense that all modular curves fall under one example, that is clearly not of modular curves. This explains why our approach to (families of) covers of the projective line can handle a barrier noted by Grothendieck to generalizing the OIT. Chap. 3, joining the 1st and last 3rd of the book, is a new approach to the lift invariant and Hurwitz space components based on the universal Frattini cover of a finite group. ChernBookOIT01-01-17.pdf

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 Z2. It then extend those techniques to Z/3 acting on Z2. 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.

Components at higher levels, corresponding to lk+1 (k>0), concatenate these two types. The HM component result generalizes the previous major result on Harbater-Mumford components. It gives a new geomtric invariant for detecting Hurwitz space components. l-adicHurReps.pdf

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 (2020).

§1 introduces Nielsen classes attached to (G,C), where C is r conjugacy classes in a finite group G. The classes support a canonical braid action. 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.

Suppose a prime l divides|G|, but G has no Z/l quotient, and C are some l' classes. Then there is a canonical tower of reduced Hurwitz spaces HG,C,l = {H(Gk,C)rd}k=0∞ with k=0 ⇔ H(G,C)rd.

The groups Gk are canonical quotients of the universal (abelianized) l-Frattini cover lψab: l~Gab → G. Its kernel interprets on the l-adic Tate module of a projective curve Jacobian ⇔ a H(G,C)rd point.

A Modular Tower (MT) is a projective sequence of components on that tower. MTs support generalizing Serre's Open Image Theorem (OIT). A Main OIT conjecture is a precise statement on the geometry of MT levels, which connect to the Regular Inverse Galois Problem applied to those groups Gk, k ≥ 1. These Main conjectures apply also to proper l-Frattini lattice quotients of lψab.

§1 uses an ''easy'' group (G=A4) to illustrate three tools by which we can compute the GQ action on the collection of MTs on the canonical Hurwitz tower:

The Branch Cycle Lemma, the Lift Invariant, and the shift-incidence matrix.

§2 introduces [Fr20], which takes on generalizing Serre's OIT: the case when G is a dihedral group Dl and C is four repetitions of the involution conjugacy class. Serre's Theorem separated decomposition groups of projective sequences of points in modular curve towers into two types: CM (complex multiplication) and GL2.

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). [Fr20] emphasizes new phenomena in cusps, and components, comparing them with modular curve towers.

On special lattice quotients these problems interpret on the canonical Hurwitz towers as unsolved problems about classical spaces (like hyperelliptic Jacobians). We look precisely at the Main OIT and RIGP conjectures on a particular family of lattice quotients, parallel to Serre's example of the full proposed result.

An expansion of this paper – Moduli Relations between l-adic representations and the Regular Inverse Galois Problem, Issue 1, Volume 5 of Graduate J. of Math. (2020), 1–76 – gives a full treatment to l-Frattini lattice quotients and l-adic representations showing the precise ties via Falting's Theorem between the RIGP and the OIT. ChernPapOITvs2-01-01-17.pdf

17. Moduli Relations between l-adic representations and the Regular Inverse Galois Problem, Issue 1, Volume 5 of Graduate J. of Math. (2020), 1–76. A preliminary paper – Introduction to Monodromy, l-adic representations and the Regular Inverse Galois Problem, – laid an historical case connecting the Regular Inverse Galois Problem (RIGP) and Serre's Open Image Theorem (OIT). Here we modernize relating the RIGP and the IGP (realizing a group G over Q) by generalizing the OIT conjectures as a precise Hilbert's Irreducibility Theorem.

Idea: For a finite group G, with conjugacy classes C which may produce regular realizations of G, it considers regular realizations of G and some groups. From this it forms a Hurwitz space tower – from a canonical Nielsen class tower – whose points correspond to regular realizations of these groups. This tower supports natural l-adic representations (l a prime dividing |G|) whose properties connect to the RIGP. Thus, the OIT generalization becomes a precise Hilbert-like Theorem.

After reviewing Nielsen class tools – with enhanced examples related to Serre's OIT – it introduces l-Frattini lattice quotients, L, attached to (G,C,l). Previously we only considered the maximal lattice – the kernel of universal (abelianized) l-Frattini cover lψab: l~Gab → G of G. Then, for L ∈ L we form a canonical tower of Hurwitz spaces HG,C.l,L = {H(Gk,C)}k=0∞ attached to (G,C,l, L). Information from L give the groups Gk. Copies of L interpret on the l-adic Tate module of a projective curve Jacobian ⇔ a level k=0 Hurwitz space point.

A Modular Tower (MT) is a projective sequence of components HG,C.l,L. A given MT, say H', has a (projective) geometric monodromy group, GH'. The Main OIT conjecture – formulated to resemble Hilbert's Theorem – compares the arithmetic monodromy groups of projective systems of points on this object to GH'. This uses the key definition/conjecture of the paper: that these be eventually l-Frattini sequences. It also uses a trick change in the Nielsen classes to make the computation of the geometric monodromy group.

Proper lattice quotients give explicit examples of Hurwitz towers on which the Main RIGP and OIT problems interpret as unsolved problems about classical spaces (starting with hyperelliptic Jacobians and concluding with our main OIT example). They relate to and benefit cases of the Torsion Conjecture on abelian varietes.

ChernPapOITvs3-03-23-20.pdf

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

l-adicReps-RIGP.html %-%-% l-adicReps-RIGP.pdf

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