1st year calculus teachers use the equation Tp(cos(ϑ))=cos(pϑ),
with Tp(w) the pth Chebychev polynomial. It is a map between complex
spheres branched over three points. I will explain why we call Tp a
dihedral function. Functions similar to it form one Mobius class:
equivalent by composing with fractional transformations.

Abel used more general dihedral Mobius classes. These form what we
now call the modular curve Y0(p). In "What Gauss Told Riemann About
Abel's Theorem" a lecture at John Thompson's 70th Birthday,
I cited Otto Neuenschwanden on the 60-year-old Gauss in conversation
with the 20-year-old Rieman. Their goal was to generalize Abel using
Gauss' harmonic functions. Riemann went far, but his early death left
an incomplete program.

To see why the generalization is non-obvious, consider: What is the
alternating (group) version of taking composites of Tp to form Tpk+1,
k 0?

This talk will use (and explain) alternating versions of modular curves
to connect two famous modern problems:

1). The Strong Torsion Conjecture (on Abelian Varieties); and

2). The Regular Inverse Galois Problem.

These spaces have cusps at points on their boundary. A cusp pairing
(the shift-incidence matrix ) helps picture these spaces. They aren't
modular curves. Still, using a result with J.P. Serre, we show how their
cusps resemble those of modular curves. That gives a version of the
renown Merel-Mazur result for these alternating spaces.