Atomic
Orbital-type cusps on Alternating Group Modular Towers
Reduced Hurwitz spaces –
spaces of r-branched Riemann
surface covers of the projective line – are dimension r-3 moduli spaces. These stacks
have cusps on their boundaries. They can have fine
moduli, but often do not. In the form of M(odular) T(ower)s they support
conjectures generalizing modular curve statements. Other researchers
use these to connect the Inverse
Galois Problem and the Strong
Torsion
Conjecture (on abelian varieties).
Like Shimura varieties – some are special cases – each MT
comes with a prime p. As many
MTs attach to p as there are p-perfect
finite groups. We get a hold on these spaces using a sh-incidence pairing on their
cusps. We will concentrate on applying the sh-incidence pairing to
infinitely many MTs where the
Main Conjectures are proved.
We chose examples of Liu and Osserman, who proved a first connectedness
result. Here the projective line covers have alternating groups as
monodromy groups, p=2 and r=4 (so tower levels are upper half
plane quotients, but not modular curves). I swear, the group theory is
surprisingly easy.
We use a "Fried-Serre" spin-lifting formula to
locate 2 cusps. Our computations were guided by the look of an atomic orbital in sh-incidence
rows. By catching 2 cusps at tower level 1 –
though there are none at level 0 – we prove the Main Conjecture. We end
by comparing p cusps in MTs with those of modular curve
towers.
This html file corresponds to §5 of the paper Connectedness
of families of sphere covers
of a given type.