Updating an Abel-Gauss-Riemann Program

Note that the pk+1th Chebychev polynomial Tpk+1 ,k ≥0, is just the composition of k+1 copies of Tp. So, the group attached to Tpk+1 is also a dihedral group extending that attached to Tp.

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.

We can directly apply the dihedral approach London1-ModCurves.pdf used on the modular curve Y0(p) to continue the Abel-Gauss-Riemann program. To see why the generalization is non-obvious, consider these questions:
  1. What would be the space attached to replacing dihedral groups, say, by alternating groups? 
  2. How can we picture these spaces, efficiently separating one from another.
  3. Then, given an appropriate prime p what is the alternating (group) version of taking composites of Tp to form Tpk+1, k ≥0? 
By using a cusp pairing – the shift-incidence matrix – we see much about these spaces. They aren't modular curves, and the tower structure starts out (at level 0) looking not at all modular curve-like. Still, using a result with J.P. Serre, we show how, at higher levels, their cusps come to resemble those of modular curves.

Sections:

§ I. More  modular curve lessons
  1. Here are the cusps, at all levels
  2. Generalize Tp compositions  Tpk+1=Tpo… oTp: Use Schur-zassenhaus and Frattini Properties 
  3. (product-one) Dp-iterations
  4. Growth of p-cusps with levels from a Spire
  5. 3 reasons to pinpoint Hurwitz space components
§ II. Modular curve-like spaces with Ans replacing Dps
Liu-Osserman: Ni((n+1)/2)4: G0=An, n ≡ 5 mod 8, p=2
  1. sh-incidence for Ni((n+1)/2)4abs,rd
  2. sh-incidence: Ni34in,rd, n=5
  3. Abs-inn sh-incidence Ni74in,rd, n=13
  4. Computing the genus of reduced spaces r=4
  5. Modular curve analog Ni(Gk,C((n+1)/2)4)in,rd and (product-one) An- iterations
  6. The Spinn-lifting invariant
  7. Listing 2-cusps in Ni(G1,C((n+1)/2)4in,rd
  8. Why an H-M Modular Tower – each level has an H-M rep. – has a spire
Appendices
  1. A modular curve-like Spire
  2. p-cusp MT Conjectures
  3. Andre's Thm. and Shimura special points