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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