Dihedral Groups: MT view of Modular curve cusps

 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 möbius class: equivalent by composing with fractional transformations. 

Abel used more general dihedral möbius classes, and these form what we now call the modular curve Y0(p). This lecture will see cusps on modular curves  from a view that generalizes to their use in Modular Towers. There are just two types of modular curve cusps: g-p', and p-cusps. The 3rd type, o-p', is missing.

Consider the modular curve X0(p): Its length p cusp is both a p-cusp and a H(arbater)-M(umford) cusp (the name first appearing in Stefan Wewers talk, and then again in Pierre Debes'). The length 1 cusp is a special g-p' cusp, the shift of the H-M cusp.

I use these facts to introduce the most important invariant of a Modular Tower level, the sh-incidence matrix. This is useful even for modular curves. It explains relations between cusps not in the traditional description because the action of the braid group doesn't appear there. 

Sections:
I. Abel and Dihedral functions
  1. The dihedral group with observations
  2. r=4 (not 3) branch dihedral functions
  3. Dragging a function by its  branch points
  4. PGL2 action; mapping class group \bar Mr
II. MT view of modular curves
  1. Classical cusp description for Γ0(pk+1)
  2. Dihedral Nielsen classes; q2 action
  3. Cusps as Cu4=<Q'', q2> orbits
  4. Cusp width principle
  5. Normalizations for listing all cusps
  6. MT  account for one absolute H4 orbit
  7. Summary: Modular curve vs all MT level cusps
Appendices:
  1. Classical Generators
  2. R(iemann)-H(urwitz)
  3. Apply R-H to MT components (r=4)
  4. Branch Cycle Argument for (G,C)
  5. sh-incidence pairing: Ni(Dp2,C24)*,rd, *=abs or in