Hurwitz Monodromy, Spin Separation and Higher Levels of a MT

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

This version has typos corrected from the in print version. A complete list of the typos is in [lum-fried0611594pap.pdf, App. C]. This paper computes everything I thought would be of interest about level one of the M(odular) T(ower) attached to A₅ and four repetitions of theconjugacy class of 3-cycles, in particular showing the Main MT Conjecture for it: No K points at high levels (K any number field). To do this required consider theory applicable to the general case. Here are results about this case.

Level 0 has genus 0, fostering much thinking about its implications for spin covers of alternating groups (documented in this paper). Level one has two components, one of genus 12, the other of genus 9. The former contains all the Harbater-Mumford representatives. All Q points (from Q realizations of the first characteristic universal 2-Frattini cover G₁ of A₅) are on the genus 12 component. So, Falting's Theorem implies there are but finitely many four branch point Q regular realizations of G₁.

Modular representation theory, and a special presentation of the degree 4 mapping class group are the main tools behind a version of Riemann-Hurwitz that helps compute the genus of Modular Tower components. It's all in the cusp geometry (as covers of the j-line) for these components. Based on results of Serre, we use Clifford algebras to interpret the group theory behind these two components. As the paper notes, for no simple group (or prime dividing its order) has there been a regular (or any other) realization of G₁over Q. So, it is an understatement that higher levels of a Modular Tower provide many challenges related to the Inverse Galois Problem.

This one example of Modular Tower analysis is really several examples, though the intense concentration is on the prime p=2 and the geometric effect of the appearance of Schur multipliers related to spin covers. Further, the cusp analysis is particularly designed to show the value of relating all cusps on the j-line covers to those attached to H-M reps.

This paper organizes some of the basics, allowing us to go after refined forms of the Main Conjecture (no rational points at high levels of the tower) for MTs of arbitrary rank [lum-fried0611594pap.pdf, §4.1.3], primes different from 2 (or p=2, but the Schur multiplier appearances don't involve spin covers), using a natural generalization of (shifts of) H-M cusps – called g-p' cusps. Further, there is a classification of Schur multipliers called Schur types, with an emphasis on those called antecedent [nilpret.pdf, §2]. This is done in the several papers following this one.