Israel Journal of Mathematics

, Volume 179, Issue 1, pp 57–125 | Cite as

Alternating groups and moduli space lifting invariants

  • Michael D. FriedEmail author


The genus of a curve discretely separates decidely different algebraic relations in two variables to focus us on the connected moduli space M g . Yet, modern applications also require a data variable (function) on the curve. The resulting spaces are versions, depending on our needs for this data variable, of Hurwitz spaces. A Nielsen class (§1.1) consists of r >- 3 conjugacy classes C in the data variable monodromy G. It generalizes the genus.

Some Nielsen classes define connected spaces. To detect, however, the components of others requires further subtler invariants. We regard our Main Result (MR) as level 0 of Spin invariant information on moduli spaces.

In the MR, G = A n (the alternating group), r counts the data variable branch points and \(C = {C_{{3^r}}}\) is r repetitions of the 3-cycle conjugacy class. This Nielsen class defines two spaces called absolute and inner: \(H{({A_n},{C_{{3^r}}})^{abs}}\) of degree n, genus g = r − (n − 1) > 0 covers and \(H{({A_n},{C_{{3^r}}})^{in}}\) parametrizing Galois closures of such covers. The parity of a spin invariant precisely identifies the two components of each space. The inner result is the deeper.

We examine the effect of combining the MR, [ArP05] and ½-canonical classes on M g . First: §5.2 considers an analog of a famous conjecture of Shafarevich: With H the composite group of all Galois extensions K/Q with group some alternating group, does the canonical map G H have pro-free kernel. Second: Thm. 6.15 produces nonzero automorphic (θ-null power) functions on the reduced Hurwitz spaces \({H_ + }{({A_n},{C_{{3^r}}})^{abs,rd}}\) (resp. \({H_ - }{({A_n},{C_{{3^r}}})^{abs,rd}}\) when r is even (resp. odd), for either g=1 or n≥12g+4.


Modulus Space Conjugacy Class Mapping Class Group Galois Extension Hyperelliptic Curve 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Hebrew University Magnes Press 2010

Authors and Affiliations

  1. 1.Emeritus, Mathematics DepartmentUC IrvineIrvineUSA

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