Introduction to Modular Towers: Generalizing dihedral group–modular curve connections:
The pdf version of the original paper modtowbeg.pdf

A finite group G is p-perfect if p divides its order but it has no Z/p quotient. This paper started the philosophy that is epitomized by the phrase: The modular curves X0(pk+1) are to the dihedral group Dp (of order 2p) as all M(odular) T(ower)s for the prime p are to any p-perfect finite group G.

HOW MTs STARTED: I put together two projects to further my 1993 application as a Senior Humboldt Research Fellow. When the fellowship came through – for it I spent about six months in Erlangen with Deiter Geyer, and then about six months at Essen with Gerhard Frey in 1994-95 – I chose the idea of Modular Towers as the more daring, riskier and ultimately more fruitful project.

To get a modern view of what has been accomplished with MTs since its inception 10 years ago see [mt-overview.html]. Still, what is from this paper is often quoted directly and has not been superceded. We list some highlights of this type.

PART I. MANY PROFINITE ASPECTS OF MODULAR CURVES: Used the dihedral modular curve connection to develop a vocabulary of inverse Galois topicswith modular curve counterparts. Especially the topic of (G,G^,C) realizations and the case of (Dp, Ap,C24) realizations  over extensions of Q of degree at most d following an argument of Frey. These discussions now model one type of MT application generalizing that for modular curves.

PART II. PRODUCTION OF THE UNIVERSAL p-FRATTINI COVER: Many explicit approaches here to the 0th characteristic p-Frattini module Mp,0 appear here. A practical use of this is to describe the module Mp,0 for p=2, 3 and 5 when G=A5 (the alternating group of degree 5). For p=2 the module consists of the D5 cosets acting on A5, modulo the sum of all cosets.

For p=5, identify  A5 with PSL2(Z/5), so the adjoint representation consists of the 3-dimensional module of trace 2x2 matrices on which PSL2(Z/5) acts by conjugation.  Then, M5,0 is the nonsplit extension of one copy of the adjoint representation by another copy of it. The basic terminology and later discussion of Loewy displays of irreducible G module subquotients on M0 come from this section.

PART III. FOUNDATIONS OF MODULAR TOWERS: The first definition of a MT occurs here. The Main Conjecture – that high levels have no K points – does not (you find that in [fried-kop97.pdf]). Much, however, of what is here is still quoted from this paper. Especially something used as a starting point by [Ca], [DDe], [DEm] and [W] (the latter was the first to read the paper, while he was a graduate student working under Frey). It is the abundant production in Thm. 3.21 of (nonempty) MTs all of whose components have definition field Q.  The criterion for this is very effective, though it essentially doesn't apply when r=4, the case that still dominates [mt-overview.html] for its direct comparison with modular curves.

What the paper calls a large lifting invariant is a GQ invariant of a MT generalizing the first appearance of the lifting invariant as it appears in Serre. Using this invariant is akin – though the geometry is different – for MTs to Shimura's profinite approach to defining level structures on systems of level sets of complex multipllication type abelian varieties.  As we have preceded to prove cases of the Main Conjecture, cases that show the need to gain calculation ability like that from [hf-can0611591.pdf, Thm. A and B], this large lifting invariant returns under the following rubric.

[hf-can0611591.pdf, Chap. 9] manages to show many precise properties of both levels 0 and 1 of many A5 MTs. Still, any general approach to proving the Main Conjecture, of a version of Serre's Open Image Theorem, etc. cannot be bound to always doing such detailed calculation. What we expect is that general principles such as those from oneorbit.pdf will assure someone with a refined interest in a particular MT level that the Main Conjecture holds. Then, a researcher can use the models of particular properties from [h4-0104289.pdf, Chap. 9] or [lum-fried0611594pap.pdf, §6].

We conclude by mentioning two such properties that take great advantage, as this paper does, of the precise description of real points on a MT that comes from [ rigRealResclass.pdf]. Both are about the (A5,C34, p=2) MT. Denote the characteristic 2-Frattini covers by {Gk=Gk(A5)}k=0∞.  Lem. 3.6  shows the spaces H(Gk,C)in all have fine moduli because the Gk's have no center. So, if there is a K point on H(Gk,C)in then there is a representing cover for that point, with K the definition field of that cover. By contrast there is an antecedent Schur multiplier central Frattini extension Rk→ Gk (kernel is Z/2). The last is similar to what [W2] produces, except his have just the not-fine-moduli property from an infinite set of different group examples, while these all come in one MT system.

[Ca]     A. Cadoret, Harbater-mumford subvarieties of moduli spaces of covers, Math. Ann. 333, No. 2 (2005), 355–391.
[DDe] P. Dèbes and B. Deschamps, Corps ψ-libres et th´eorie inverse de Galois infinie, J. fũr die reine und angew. Math. 574 (2004), 197–218
[DEm] P. Dèbes and M. Emsalem, Harbater-Mumford Components and Hurwitz Towers, J. Inst. of Mathematics of Jussieu (5/03, 2005), 351–371.
[Se]     J.P. Serre, Relˆevements dans A˜n, C. R. Acad. Sci. Paris 311 (1990), 477–482.
[W]      S. Wewers, Construction of Hurwitz spaces, Thesis, Institut f¨ũr Experimentelle Mathematik 21 (1998), 1–79.
[W2]    S. Wewers, Fields of moduli and field of definition of Galois covers, same volume as [h4-0104289.pdf], 221–245.

Mike Fried 02/08/2007 mfri4@aol.com mfried@math.uci.edu