Hurwitz monodromy, spin separation and higher levels of a Modular Tower

By P. Bailey, M. Fried. 2001

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Each finite p-perfect group G (p a prime) has a universal central p-extension coming from the p part of its Schur multiplier. Serre gave a Stiefel- Whitney class approach to analyzing spin covers of alternating groups (p = 2) aimed at geometric covering space problems that included their regular realization for the Inverse Galois Problem. A special case of a general result is that any finite simple group with a nontrivial p part to its Schur multiplier 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 Modular Towers, capture the geometry of these many appearances of Schur multipliers in degeneration phenomena of Harbater-Mumford cover representatives. These are modular curve tower generalizations. So, they inspire conjectures akin to Serre's open image theorem, including 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 θ-nulls on these moduli spaces. The results immedi- ately 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.

Subfields: Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/64108c5f85b6a51776fd8e6b401d4674f59ed3e1