Mike Fried, Emeritus, UC Irvine

How Pure-cycle Nielsen classes test the Main Modular Tower Conjecture: Full talk file rims-fried10-26-06.pdf


Abstract: Modular curves, like X0(pk+1), are moduli spaces of genus 0 covers. As k varies they form a tower. Level k points represent rational functions f: P1w---> P1z with branch points z1,..,z4 having local monodromy of order 2, and the dihedral group Dpk+1 as monodromy group. We view each level k point as a set: µ¤f¤µ' with µ and µ'  running over Mobius transformations.

Our Hurwitz space notation H(Dpk+1,C)abs,rd for X0(pk+1) hints at more generality: H(G,C)*. Here G is a finite group, C are conjugacy classes in G, and * is an equivalence relation (*=abs,rd: absolute-reduced above).

Clebsch in the 1870s used the simple branching case: G=Sn; conjugacy classes are r ≥ 4 repetitions of the 2-cycle class). I use them like this: To solve a problem, decipher existence of certain types of covers from moduli space properties. Figuring connected components starts the geometry.

Talk Introduction: We will employ the solution of Serre's problem on spin covers of alternating groups with 3-cycles [Fr06b], to give a graphic introduction -- using the sh-incidence matrix -- to the Fried-Serre lifting invariant (generalizing spin structures).

Talk Part 1. The overlap of this 3-cycles result with recent work of Liu-Osserman suggests a simultaneous generalization of both to pure-cycle Nielsen classes (one disjoint cycle in the cycle-type). By applying the lifting invariant we can guess the precise form of that generalization. If true, it  fulfills an explicit form of Conway-Fried-Parker-Voelklein result on describing components of Hurwitz spaces. Combining the lifting invariant and the Branch-Cycle-Lemma would then tell us precisely about these pure-cycle Hurwitz space components and their definition fields.

Talk Part 2. First a brief statement on Modular Towers: For the prime p and any p' conjugacy classes they are to modular curve towers (for p) as all p-perfect groups are to the dihedral group [D06] Dp. I will apply the Liu-Osserman odd-order conjugacy case when r=4 to the Main Conjecture on Modular Towers [LO06].

The Main Conjecture: High Tower levels have no rational points -- long known for modular curve towers. Cadoret recently showed the S(trong) T(orsion) Conjecture on abelian varieties implies it [Ca05]. In turn, the Main Conjecture says what we expect for the R(egular) I(nverse) G(alois) P(roblem) generalizes the famous Mazur-Merel result.

We'll show the Main Conjecture translates to high tower levels having p cusps. We can often compute yes or no from the lifting invariant, using the p-Poincare duality result of Weigel ([Se06] and [W05]). This makes properties of the lifting invariant a serious test for the STC.

[C05] A. Cadoret, Rational points on Hurwitz towers, preprint as of Jan.~2006, 1–30.

[D06] P. Debes, Modular Towers: Construction and Diophantine Questions, same vol.~as [Fr06a].
[Fr06a] 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),  Seminaires et Congres, 13, (2006), 165–233.

lum03-12-04.html has related talk and pdf files.

[Fr06b] M.D. Fried, Alternating groups and moduli space lifting invariants, to appear in Israel J. 2009, 1–46.

[LO06] F. Liu and B. Osserman, The Irreducibility of certain pure-cycle Hurwitz spaces, preprint as of August, 2006.

[Se06] D. Semmen, Modular Representations for Modular Towers same vol.~as [Fr06a].
[S90] J.P. Serre, Jean-Pierre Serre, Relèvements dans Ȁn, C. R. Acad. Sci.  Paris Sèr. I Math. 311 (1990), no. 8, 477--482. MR1076476 (91m:20010)

[W05] 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.


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