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(p^k+1), are moduli spaces of genus 0 covers. As k varies they form a tower. Level k points represent rational functions f: P¹_w---> P¹_z with branch points z₁,..,z₄ having local monodromy of order 2, and the dihedral group D_p^k+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(D_p^k+1,C)^abs,rd for X₀(p^k+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=S_n; 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 one component.

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 D_p.

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.