Modular Tower Time Line

This file consists of a list of papers (by year) for particular events in the development of M(odular)T(ower)s. I've divided these into three periods.
Each item connects to a fuller explanation of the history and significance of the contribution. Three html files provide handy reminders on basics guiding progress on M(odular) T(ower)s. We refer to sections in them.
1. The R(egular) I(nverse) G(alois) P(roblem), its literature and relation to Nielsen classes and the MT conjectures: RIGP.html.
2. Nielsen classes, a genus generalization that separates sphere covers into recognizable types. Denoted Ni(G,C), for conjugacy classes C=C1,…,Cr of a finite group G. The set consists of generating r-tuples (g1,…,gr), in (some order in) the classes of C, satisfying the product-one condition g1gr=1. R(iemann)-H(urwitz) gives the corresponding sphere cover genus.
3. The B(ranch) C(ycle) L(emma) tying definition fields of covers (and their automorphisms) to branch point locations: Branch-Cycle-Lem.html
These files let us speak of progress on MTs without great elaboration. Other such files, building on the three above, explain more what the papers are about. Finally, this page ends with further references that give results technically related to MTs in the abstracts to papers.

Individual MTs have an attached prime (usually denoted p). When the MT data passes a lift invariant test (coming up in [Se90a]), then the MT is an infinite (projective) system of nonempty levels (result stated precisely in [Fr06] below). Each level is a normal algebraic variety, all levels covering the classical j-line when r=4, and an r-3 dimensional generalization of it for larger r. Indeed, the levels are reduced Hurwitz spaces. Most modern applications of algebraic equations requires more data than is given by the moduli of curves of genus, or even of Shimura varieties. Hurwitz spaces, however, do carry such data, and unlike rougher use of curve families, they retain the virtue of having moduli properties.

MTs come with what we call the usual MT conditions: Each has an attached group G, and a collection of r conjugacy classes, C, in G with p' elements: of orders prime to p. Further, G is p-perfect: p divides its order, but G has no surjective homomorphism to the cyclic group Z/p. For G a dihedral group, with p odd and r=4, we are in the case of modular curve towers. So, MTs generalizes modular curves towers. Since there are so many p-perfect groups, the generalization is huge.

The Main Conjectures are these: High tower levels have general type; and even if all levels have definition field a finite extension K of Q, still K points disappear (off the cusps) at high levels. Bringing particular MTs alive plays on cusps, as do modular curves. Part III of our TimeLine includes precise comparison of MT cusps with those of modular curve towers, consequences of this, and two different methods that have given substantial progress on the Main Conjectures.

The intimacy between the Inverse Galois Problem and MTs gives an elementary urgency to identifying their levels. A graphical device – the sh(ift)-incidence matrix – displays these cusps, despite only for modular curves having congruence subgroups at our disposal.

Shimura varieties are another generalization of modular curves. They also have towers, and primes, etc. The connection of abelian varieties to MTs has been made in several ways. There is one easily stated standout: The S(trong) T(orsion) C(onjecture) on torsion points on abelian varieties implies the rational point conjecture on MTs (see [CaDe08].

It is, however, by labeling MT cusps that we see tools for generalizing Serre's Open Image Theorem, especially through recognizing p-Frattini covers and using reduced Hurwitz spaces (defining tower levels). These are examples that show my web site has help for related topics.

I. Lessons from Dihedral groups – Before '95:

This section goes from well-known projects to their connection with the MT program. Its reference to [DeFr94] – which included e-mail exchanges with Mazur – sets the stage for the recent-years' division of the project into two branches. The arithmetic has concentrated in the hands of Pierre Dèbes and his collaborators Cadoret, Deschamps and Emsalem. The structure of particular MTs – based on homological algebra and the geometry of the spaces (cusps and components) – follows my papers and my relation to Bailey, Kopeliovic, Liu-Osserman, Semmen, Serre and Weigel. The effects of quoted work of Ihara, Matsumoto and Wewers, all present at my first talks on MTs, is harder to classify.

[Sh64] G. Shimura, Abelian Varieties with Complex Multiplication and Modular Functions, Princeton U. Press, 1964 (with Y. Taniyama), latest edition 1998. ✺ I studied this source during my two year post-doctoral 67-69 at IAS. Standout observations: Relating a moduli space's properties to objects represented by its points, through the Weil co-cycle condition, especially in canonically finding the definition fields of a tower.

Results from it: The Branch-Cycle-Lem.html and its early uses (dav-red.pdf, UMShortStory.html and HurwMonGG.pdf for problems not previously considered as moduli-related. Also, a model for producing automorphic functions on Hurwitz spaces.

[Se68] J.P. Serre, Abelian l-adic representations and elliptic curves, New York, Benj. Publ., 1968. ✺ That the monodromy groups of towers of modular curve covers have a Frattini property, suggesting the general expectation for the action of GQ on a projective system of points over a j-line point in Q. Use of the p cusps on modular curves, and their attachment to Tate's p-adically uniformized elliptic curves to decipher the GQ action when j is p-adically "close to" ∞. For K a complex quadratic extension of Q, the technical point of complex multiplication is the discussion of 1-dimensional characters of GK on the Qp vector space – Tate module, or 1st p-adic étale cohomomology – of an elliptic curve with complex multiplication by K. On the 2nd p-adic étale cohomomology it is the cyclotomic character, while on the 1st there is no subrepresentation of any power of the cyclotomic character. Roughly: A small part of Abelian extensions of K are cyclotomic, a result that generalizes to higher dimensional complex multiplication in [Sh64]. As [Ri90] emphasizes Serre's book is still relevant, especially for the role of abelian characters, those represented by actions on Tate modules (from abelian varieties), and those not.

Results from it: Recognition that the modular curve Frattini property is inherited from a Frattini property for sequences of dihedral groups; so the proper generalization of Serre's main result to MTs should exploit this [Fr06, ?6.3]. The full (and comfortable) completion of Serre's O(pen) I(mage) T(heorem) awaited replacement of an unpublished Tate piece by ingredients from Falting's Thm. [Fa83] (as in [Se97b]).

[Fr78] M. D. Fried, Galois groups and complex multiplication, Trans. Amer. Math. Soc. 235 (1978), 141–163. MR MR472917 (81c:14015) ✺ Identification of modular curves with Hurwitz spaces, and the classification of the Schur-conjecture for rational functions as equivalent to part of [Se68]. Generalizing the Inverse Galois Problem, to geometric-arithmetic realizations, using Hurwitz spaces.

Down-to-Earth result from it: From the GL2 part of the OIT, explicit production from each elliptic curve over Q with non-integral j-invariant, infinitely many primes p and a degree p2 rational function f decomposing, over the algebraic closure, into two rational functions of degree p, with no such decomposition over Q ([GMS03] and [Fr05, Prop. 6.6]). Abstract results from it: Over suitable fields, you can achieve any geometric-arithmetic realizations, giving the first proven group presentation of GQ. Then, formulating the likely generalization of Shafarevich's Cyclotomic Conjecture GQpresentation.pdf.

[Ih86] Y. Ihara, Profinite braid groups, Galois representations and complex multiplications, Ann. of Math. (2) 123 (1986), no. 1, 43–106. MR MR825839 (87c:11055) ✺ The similar titles with [Fr78] gives away the similar influence of Shimura. Both played on using and interpreting braid group actions, a monodromy action capturing data from curves, versus from abelian varieties. A moduli interpretation of "complex multiplications" required to generate the field extension giving the second commutator of GQ.

Down-to-Earth result from it: Generating the second commutator (arithmetic) extensions using Jacobi sums derived from Fermat curves. Abstract result from it: An interpretation of Grothendieck-Teichm?ller on towers of Hurwitz spaces [IM95].

[Se90a] J.P. Serre, Relèvements dans Ãn, C. R. Acad. Sci. Paris 311 (1990), 477–482. ✺ This suggested a general context for viewing mysterious and previously inaccessible central Frattini extensions of groups, yielding to the braid technique – in this case a formula for deciding if a regular realization of An extends to the Spin cover Spinn (what Serre calls Ãn) of An. A braid orbit O in Ni(An,C), with C of odd-order elements, passes the (spin) lift invariant test if the natural (one-one) map Ni(Spinn,C) → Ni(An,C) maps onto O. Main Result: If the genus attached to Ni(An,C) is 0, then the test depends only on the Nielsen class and not on O.

Results inspired by it: Formulation of the main connectedness result on Hurwitz spaces CFPV.html. Classification and application of Frattini central extensions of centerless groups [Fr02, § 3 and 4].

[Se90b] J.-P. Serre, Revètements à ramification impaire et thèta-caractèristiques, C. R. Acad. Sci. Paris 311 (1990), 547–552. ✺ Example result: A formula for the parity of a uniquely defined half-canonical class on any odd-branched Riemann surface cover of the sphere. It is the sum mod 2 of an invariant depending only on the Nielsen class of the cover, and the spin lift invariant mentioned in [Se90a].

Result from it: Production of Hurwitz-Torelli automorphic functions on specific Hurwitz spaces through the production of even theta-nulls [Fr09a, § 6.2].

[DeFr94] P. Dèbes and M. D. Fried, Nonrigid situations in constructive Galois theory, Pacific Journal 163(1994), 81–122. ✺ Example result: Formulation of the Main MT conjecture for dihedral groups like this. Suppose, for some prime p, there are Q regular realizations of all the dihedral groups {Dpk}k=0 with some bound r0 on their number of branch points. Then (equivalently), the Branch-Cycle-lemma implies there is an even integer r1 (≤ r0) and for each k, there is a dimension (r1-2)/2 hyperelliptic Jacobian (over Q) with a μ(pk) point for each k (≥ 0). The Involution Realization Conjecture says the last is impossible: There is a uniform bound as n varies on μ(n) torsion points on hyperelliptic Jacobians of a fixed dimension, over any given number field. (The only proven case, r1=4, is the Mazur-Merel result bounding torsion on elliptic curves.) If a subrepresentation of the cyclotomic character occurred on the p-Tate module of a hyperelliptic Jacobian (see [Se68]), the Involution Realization Conjecture would be blatantly false.

Result from it: Formulation of the general Main MT conjecture [FrKop97]. Still missing result: Find μ(n) torsion points on any hyperelliptic Jacobian for all – even infinitely many – s.

II. Construction and Main Conjectures on MTs – '95–'04:

The extension of a p-perfect finite group G called its universal p-Frattini cover has kernel a pro-free pro-p group of finite rank. Mod out by the kernel's commutator subgroup to get its abelianization. Both extensions have characteristic series of quotients, respectively, ΓG,p= {Gk}k=0 and ΓG,p,ab ={Gk,ab}k=0. We can use either series to define a tower of Hurwitz spaces.

Either finding rational points on tower levels, or proving they don't exist, makes sense whichever series we use. Clearly, however, it is more challenging to find rational points on level k defined from ΓG,p. Similarly, it is more affirmative of the Main Conjecture, if we find there are no rational points on some level defined from ΓG,p,ab. That is how the progress appears below. Results supporting the Main MT conjectures are about the abelianized towers. Results providing points on tower levels are for the series ΓG,p defining the full tower.

[Fr95] M. D. Fried, Introduction to Modular Towers: Generalizing dihedral  group–modular curve connections, Recent Developments in the Inverse Galois Problem, Cont. Math., proceedings ofAMS-NSF  Summer Conference 1994, Seattle 186(1995),111–171. ✺ Formulation of the MT levels, based on the characteristic quotients of the universal p-Frattini cover of a finite group. Properties of the universal p-Frattini cover that translate to fine moduli. A generalization of the lift invariant, proof that it is a braid invariant, and how it acts under the absolute Galois group. A criterion for finding Q components – called Harbater-Mumford – of Hurwitz spaces, so guaranteeing existence of projective sequences of absolutely irreducible Q components on MTs, beyond those of modular curves.

[FrKop97] M. D. Fried and Y. 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. ✺ Suppose G is a group with many known regular realizations; maybe An semidirect product some finite abelian group: see RIGP.html, § IV.1. Consider, for some prime p for which G is p-perfect if there are regular realizations of the whole series of ΓG,p,ab over some number field K. Thm. 4.4 asks this under the condition that all such have a uniform bound on the number of branch points – saying nothing about which conjugacy classes we use.

Conclusion: Such regular realizations are only possible by restricting to p' conjugacy classes. Even then, there must exist a MT over K with a K point at every level. This geometric Fontaine-Mazur analog [Fr06b], generalizes for each such G the Involution Realization Conjecture [DeFr94] for dihedral groups.

[We98] S. Wewers, Construction of Hurwitz spaces, Thesis, Institut für Experimentelle Mathematik 21 (1998), 1–79. ✺ Develops a compactification of Hurwitz spaces based on the stable-compactification theorem. Allows a standard comparison – contrasting with the group theoretic use of specialization sequences in [Fr95, Thm. 3.21] – for labeling Harbater-Mumford cusps as lying on Harbater-Mumford components. Both compactifications are compatible with the MT construction (they form natural projective systems), and they support that only for primes dividing |G| can the system have bad reduction.

[BaFr02] P. Bailey and M. D. Fried, 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 of possible comparison with modular curves about level one (and level 0) of the M(odular) T(ower) attached to A5 and four repetitions of the conjugacy class of 3-cycles. Shows 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, of genus 12 and genus 9. Concludes with a conceptual accounting of all cusps, and all real points on the whole tower (none on the genus 9 component).

Visualization of cusps on both levels uses a new pairing on them, based on Nielsen classes: The sh(ift)-incidence matrix (§ 4 of CSCshInc.pdf has many examples, including level 1 of modular curves). A version of the spin cover (extending the domain of use of [Se90a]) obstructs anything beyond level 1 for the genus 9 component. Much is made of this argument: Any prime l of good reduction, for which there are Z/l points at each level of a MT, would automatically give the trivial power of the cyclotomic character acting on a Tate module, as disallowed in [Se68].

[Fr02] M. D. Fried, Moduli of relatively nilpotent extensions, Communications in Arithmetic Fundamental Group, Inst. of Math. Science Analysis, vol. 1267, RIMS, Kyoto, Japan, 2002, pp. 70–94. ✺ Moduli of relatively nilpotent extensions, Inst. of Math. Science Analysis 1267, June 2002, Communications in Arithmetic Fundamental Groups, 70–94. ✺ Gives a precise description of the p-Frattini module for any p-perfect G (Thm. 2.8), and therefore of the sequence ΓG,p,ab. § 4 labels Schur multiplier types, especially those called antecedent. Example: In MTs where G=An, the antecedent to the level 0 spin cover affects MT components and cusps at all levels ≥ 1 (see [Fr06]).

[Sem02] D. Semmen, The Frattini module and p'-automorphisms of free pro-p groups, Comm. in Arith. Fund. Groups, Inst. Math/Sci Analysis 1267 (2002), Kyoto University, RIMS (2002), 177–188. ✺ Striking challenges to the Inverse Galois problem arise by using any one p-perfect group, and analyzing characteristic p-Frattini extensions and the components of their corresponding Hurwitz spaces. In lieu of the CFPV.html result and [Fr95, Thm. 3.21], the most serious phenomenon in unexplained Hurwitz space components – making it difficult to identify definition fields – comes from nonbraidable outer automorphisms of groups. Such have occurred at several level 1 MTs, producing two separate Harbater-Mumford components.

Here are techniques for computing p-Frattini extension outer automorphisms. Then, in cases from [BaFr02, ? 9] (especially where G=A4, p=2 and the reduced Hurwitz space components have genus 1) it identifies the non-braidable outer automorphism.

[DeDes04] P. Dèbes and B. Deschamps, Corps ψ-libres et thèorie inverse de Galois infinie, J. für die reine und angew. Math. 574 (2004), 197–218. ✺ If the arithmetic Main MT were wrong, then there would be a finite group G satisfying the usual conditions for p and C so that for some number field K, the corresponding MT would have a K point at every level. Using compactifications of the MT levels (as in [We98]), for almost all primes l of K, this would give a projective system of OK,l (integers of K completed at l) points on cusps. The Main Results here considered what MTs (and some generalizations) would support ZK,l points for almost all l (inverse to [BaFr02]) using Harbater patching.

III. Progress on the Main MT Conjectures – '05–'09:

As with modular curves, the actual MT levels come alive by recognizing moduli properties attached to particular (sequences) of cusps. It often happens with MTs that level 0 of the tower has no resemblance to modular curves, though a modular curve resemblance arises at higher levels.

Level 0 of alternating group towers illustrate: They have little resemblance to modular curves. Yet, often level 1 starts a subtree of cusps that contains the cusptree of modular curves. We can see this from a (preliminary) classification of cusps [Fr06, ?3]. By striving for appropriate generalizations of Serre's O(pen) I(mage) T(heorem) to MTs, present MT projects are entwining the general theory of abelian varieties with properties of finite simple groups.

[We05] T. Weigel, Maximal l-frattini quotients of l-poincare duality groups of dimension 2, volume for O.H. Kegel on his 70th birthday, Arkiv der Mathematik--Basel, 2005. ✺ [Se97a, I.4.5] extends the classical notion of Poincar? duality to any pro-p group. Especially it was applied to the pro-p completion of the fundamental group of a compact Riemann surface of any given genus. This paper uses the extended notion, intended for groups that have pro-p groups as extensions of finite groups. Main Result: The p-Frattini cover ΓG,p (and ΓG,p,ab) is a p-Poincar? duality group of dimension 2.

Result from it [Fr06, Cor. 4.19] (compare with the statement in [Se90a]): Assume the usual MT conditions, and let RG,p be the maximal p central Frattini extension of G. Then, there is a (nonempty) abelianized MT over the Hurwitz space component corresponding to O if and only if the natural (one-one) map – compare with [Se90a] – Ni(RG,p,C) → Ni(G,C) is onto O.

[DeEm05] P. Dèbes and M. Emsalem, Harbater-Mumford Components and Hurwitz Towers, J. Inst. of Mathematics of Jussieu (5/03, 2005), 351–371. ✺ Continuing the results of [DeDes04], based on [We98], ties together the notions of Harbater_Mumford components and the points on cusps that correspond to them, connecting several threads in the theory. As an application, they construct, for every projective system Gn, a tower of corresponding Hurwitz spaces, geometrically irreducible and defined over Q (using the criterion of [Fr95, Thm. 3.21]), which admits projective systems of points over the Witt vectors with algebraically closed residue field of Zp, avoiding only those p dividing some |Gn|.

Applied to MTs and the sequence ΓG,p, the results are much stronger. This is done explicitly using Harbater-Mumford cusps (as in [We98]), with the primes dividing |G| and the cyclotomic extension defined by the orders of elements in C to consider. [Fr06, Fratt. Princ. 2] says existence of a g-p' cusp defines a regular realization of ΓG,p over any algebraic closure of Q in the Nielsen class, and likely this is if and only if. The approach to more precise results has been to consider a Harbater patching converse: Identify the type of a g-p' cusp that supports a Witt-vector realization of ΓG,p.

[De06] P. Dèbes, Modular towers: Construction and diophantine questions, Luminy Conference on Arithmetic and Geometric Galois Theory), vol. 13, Seminaire et Congres, 2006. ✺ This exposition starts with expositions on [DeFr94], [FrKop97, [DeDes04] and [DeEm05]. One worthy goal would replace the use of full Hurwitz spaces in a tower with lower dimension, maybe even 1, spaces with the potential of giving regular realizations of some of the groups in, say, the series ΓG,p. For this there is the result of Cadoret [Ca06]: Such new varieties – curves or surfaces – are obtained as subvarieties of the HM-components by specializing all branch points but one or two. To preserve irreducibility requires an (intricate) transitivity condition of some braid action, achievable with a specific list of restrictions by some groups G.

[Fr06] M. D. Fried, 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. Dèbes), Sem. et Congres, Vol. 13 (2006), 165–233. ✺ Cusp types and Cusp tree on a Modular Tower: If you compactify the tower levels, you get complete spaces, with cusps lying on their boundary. The MT approach allows identifying these cusps using elementary finite group theory. They are of three types (? 3.2.1): p-cusps, g(roup)-p' and o(nly)-p'. Modular curve towers have only the first two types, with the g-p' cusps on them the special kind called shifts of H(arbater)-M(umford). Let O be a braid orbit on Ni(G,C).
1. There is a full MT over the Hurwitz space component corresponding to O if O contains a g-p' representative (no need to check central Frattini extensions).
2. When r=4, there is a small list of possibilities for MTs that could fail the Main Conjectures (Thm. 5.1). Example modular curve generalization: If O contains a H-M cusp that is also a p-cusp, the Main Conjectures hold explicitly for any MT over O.
3. Generalizing the precise MT criterion given in [We05, Princ. 4.23] gives a lift invariant criterion for p-cusps to lie above o-p' cusps.
[LO08] F. Liu and B. Osserman, The Irreducibility of Certain Pure-cycle Hurwitz Spaces, Amer. J. Math. # 6, vol. 130 (2008), 1687–1708. ✺ Showed the absolute Hurwitz spaces of pure-cycle (elements in the conjugacy class have only one length ≥ 2 disjoint cycle) genus 0 covers have one connected component. There is a conspicuous overlap with the 3-cycle result of [Fr09a, Thm. 1.3], the case of four 3-cycles in A5. The impression from [LO08; §5] is that all these Hurwitz spaces are similar, without significant distinguishing properties. [Fr09b, §5], however, dispels this impression. First by noting that subsets of these Nielsen classes can have differing inner Hurwitz spaces, varying in having one or two components. Then, by detecting seriously diverging behaviors in their level 1 cusps.

[CaDe08] A. Cadoret and P. Dèbes, Abelian obstructions in inverse Galois theory, Manuscripta Mathematica, 128/3 (2009), 329–341. ✺ If a finite group G has a regular realization over Q, then the abelianization of its p-Sylow subgroups has order (pu) bounded by an expression in their index m in G, the branch point number r and the smallest prime l of good reduction of the cover. This is a new constraint for the regular inverse Galois problem. To whit: If pu is large compared to r and m, the covers branch points must coalesce modulo some prime l; an l-adic measure of proximity to a cusp on the corresponding Hurwitz space.

A striking conjecture: Some expression in r and m, independent of l, bounds pu. This follows from the S(trong) T(orsion) C(onjecture) on abelian varieties, and it gives forms of the Main MT conjecture.

[CaTa09] A. Cadoret and A. Tamagawa, Uniform boundedness of p-primary torsion of Abelian Schemes, preprint as of June 2008. ✺ Let χ: GKZp* be a character, and A[p](χ) the p-torsion on an abelian variety A on which the action is through χ-multiplication. Assume χ does not appear as a subrepresentation on any Tate module of any abelian variety (see [Se68, [DeFr94] and [BaFr02]). Then, for A varying in a 1-dimensional family over a curve S defined over K, there is a uniform bound on |As[p](χ)| for sS(K). In particular, this gives the Main MT conjecture when r=4. Further observations:
1. The §5.2 result says: If you have a p-Frattini cover of G with kernel having Zp as a quotient, then the corresponding version of a MT has a level with no K points. Examples of [Fr06, §6.3] show that whether or not this applies to the whole abelianized p-Frattini tower depends on G.
2. [Fr09b, Prop. 5.15] displays a MT level where the genus exceeds 1. So, [Fa83] implies this level has, for any K, but finitely many rational points. While more explicit than this §5.2 result, ultimate bounds for either method depend on conjectures like those in [CaDe08].
[Fr09a] M. D. Fried,  Alternating groups and moduli space lifting Invariants, description and properties of spaces of 3-cycle covers, Arxiv #0611591. 01/04/09 To appear in Israel J. ✺ The paper's main theorem strengthens a theorem of Fried-Serre on deciding when sphere covers with odd-order branching lift to unramified Spin covers. Each component of such a Hurwitz space carries a canonical half-canonical class. In many cases, including the 3-cycle cases, it uses [Se90b] to separate components precisely according to the evenness or oddness of these half-canonical classes. Example corollaries: To produce Hurwitz-Torelli automorphic functions on Hurwitz spaces, and to draw Inverse Galois Conclusions (see §4 of RIGP.html).

[Fr09b] M. D. Fried, Connectedness of families of sphere covers of An-type, Preprint as of June 2008. ✺ Gives stronger results in Liu-Osserman cases, by considering the inner (rather than absolute) Hurwitz spaces. Prop. 5.15 uses the sh-incidence matrix to display cusps, elliptic fixed points, and and genuses of the inner Hurwitz spaces in two infinite lists of [LO08] examples. (? 4 of CSCshInc.pdf compares these examples with applying sh-incidence to level 1 of modular curve towers.) In one there are two level 0 components (conjugate over a quadratic extension of Q), and for the other just one. Further, applying #3 of [Fr06], the nature of the 2-cusps in the MTs over them are very different. None have 2-cusps at level 0. For the list with level 0 connected, the tree of cusps, starting at level 1, contains a subtree isomorphic to the cusp tree on a modular curve tower: it has a spire. For the other list, there are 2-cusps, though not quite like those of modular curves.

A spire makes plausible a version of Serre's O(pen) I(mage) T(heorem) on such MTs. The goal of recognizing p-cusps is a key to showing high MT levels have general type, and gives an approach to the Main Conjectures for r≥ 5.