Mike Fried, UC Irvine and MSU-Billings
Title: Connectedness of moduli
spaces of Riemann Surface covers
Abstract: Clebsh in the
1870s applied Hurwitz spaces to the moduli of curves of genus g, though it goes back to Abel for
some modular curves. 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
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).
Many know Clebsch 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.
Two applications feature two crucial tools.
Applications: Davenport's problem on polynomial values over finite
fields; and Serre's problem on spin covers of alternating groups with
3-cycles. The Hurwitz spaces I use in both cases usually have more than
Tools: The Fried-Serre lifting invariant (generalizing spin
structures); and The Branch Cycle Lemma.
The overlap of the 3-cycles result with recent work of Liu-Osserman
suggests a simultaneous generalization of both. I will apply the
Liu-Osserman odd-order conjugacy case to the Main Conjecture on Modular
Towers. My briefest 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 Dp.
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. In turn, the Main
Conjecture implies the R(egular) I(nverse) G(alois) P(roblem)
generalizes the famous Mazur-Merel result.
We take the Liu-Osserman case where all Modular Tower levels are j-line covers (but not modular
curves). 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. This makes properties of
the lifting invariant a serious
test for the STC.