Hurwitz monodromy, spin separation and higher levels of a modular tower

Details Hurwitz monodromy, spin separation and higher levels of a modular tower Hurwitz monodromy, spin separation and higher levels of a modular tower

2001-04-18

arxiv.org/abs/math/0104289v2

in Proceedings of Symposia in Pure Mathematics {\bf 70} (2002) editors M. Fried and Y. Ihara, 1999 von Neumann Symposium on Ar

Mathematics

Number Theory

Scientific paper

Each finite $p$-perfect group $G$ ($p$ a prime) has a universal central $p$-extension. For a perfect group these central extensions come from its {\sl Schur multiplier}. Serre gave a Stiefel-Whitney class approach to analyzing spin covers of alternating groups ($p=2$) aimed at geometric covering space problems. This included the regular version of the Inverse Galois Problem. Every finite simple group with order divisible by $p$ has an infinite string of perfect centerless group covers exhibiting nontrivial Schur multipliers for the prime $p$. Sequences of moduli spaces of curves attached to $G$ and $p$, called {\sl Modular Towers}, capture the geometry of these many appearances of Schur multipliers in degeneration phenomena of {\sl Harbater-Mumford cover representatives}. These modular curve tower generalizations inspire conjectures akin to Serre's open image theorem. This includes that at suitably high levels we expect no rational points. Guided by two papers of Serre's, these cases reveal common appearance of spin structures producing $\theta$-nulls on these moduli spaces. The results immediately apply to all the expected Inverse Galois topics. This includes systematic exposure of moduli spaces having points where the field of moduli is a field of definition and other points where it is not.

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