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othlist-mt: Arithmetic and Homological Contributors to Modular Towers: Ties to the article called Modular Tower Time Line For an html and pdf (or ppt) file with the same name, the html is an exposition. Click on any of the [ 12] items below.
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The next list is of Abstracts of Contributors to Modular Towers: Each abstract starts with a paper designator, [NmYr] that matches the same designator in a fuller discussion connecting these papers to the Modular Tower Time Line. Things like the lift invariant and examples of Nielsen classes have brief discussions in the preludes to each of the three sections of that document. Papers are chronological, and I've put URLs to related discussions. The notations ΓG,p and ΓG,p,ab are, respectively, for the full (resp. abelianized) universal p-Frattini cover of the finite group G. Realization (resp. Main Conjecture) results are stronger with the former (resp. latter) as explained in §III of the Time Line.

Rainbow Line

✺ [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 (the Nielsen class) 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 this 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]. serre-oddraminv.pdf

✺ [Se90b] J.-P. Serre, Revêtements à ramification impaire et thêta-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]. serre-oddramtheta.pdf

✺ [Ri90] K. Ribet, Review of Abelian l-adic representations and elliptic curves by J-P. Serre. Bull. Amer. Math. Soc. (N.S.) 22 (1990), no. 1, 214–218. This reference is tied to the discussion of Se68 to explain the use made of various 1-dimensional characters of the absolute Galois group GQ of Q – in [DeFr94] and [CaTa09] – that do or do not appear on Abelian varieties.
RibetRevSerreL-adic90.pdf

✺ [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 (order divisible by p, but having no Z/p quotient), 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. SemmenpFrattiniOutAuto.pdf

✺ [DeDes04] P. Dèbes and B. Deschamps, Corps ψ-libres et théorie 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.
DeDesCorp-psi_L.pdf

✺ [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}n=0∞, 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 full universal p-Frattini sequence ΓG,p (see the Time Line comment), the results are much stronger. This is done explicitly using Harbater-Mumford cusps (as in [W98]), 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. deb-emsHM.pdf

✺ [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 Poincaré 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-Poincaré duality group of dimension 2. weig-p-Poincaredual.pdf

✺ [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. lum-debes09-05-06-pap.pdf

✺ [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 intuition. 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. hurwitzLiu-Oss.pdf

✺ [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 Modular Tower conjecture. Cad-DebAbelConstIG.pdf

✺ [CaTa09] A. Cadoret and A. Tamagawa, Uniform boundedness of p-primary torsion of Abelian Schemes, preprint as of June 2008. ✺ Let χ: GK → Zp* 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 s ∈ S(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].
Cad-TamBoundTorChar0.pdf

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