Notes for Oberwolfach talk, April 15 to 21


0.  Result is singular if 2 coordinates, wk'  wl' both ramify over the same branch point zi .

1. Fiber product construction of Galois closure: Get arithmetic monodromy by taking union of components conjugate over K. 

2. Covers in a Nielsen class, all of same genus Nielsen class generalizes moduli of genus g curves. 

3. Whole construction is in excruciating detail on my website, as Chapter 4 of Riemann's Existence Theorem. It has been there since 1980, and it is referenced often by others. 


4a. The element R = q1 · · · qr−1qr−1 · · · q1  applied to g sends it to the conjugate of g  by g1 . But for  groups with generators and relations,  mod out by all conjugates of R; running over qi Rqi-1 applied to g you easily see that the minimal equivalence relation for H_r action is inner. 

4b. The natural map from inner classes to absolute classes (most classical applications belong here) comes  from the natural map from inner Hurwitz spaces to absolute Hurwitz spaces. Famous classical example:  map between the modular curves  X1(lk+1) X0(lk+1) when both are interpreted as moduli spaces of dihedral group covers. 


5. The expression γ0 γ1 γ∞ =1 using  q1=q3, and the braid relation from writing R as: 

 (q1q2q1)(q1q2q1)= q2(q1q2)sh, also (q1q2)q1(q1q2q1) =1 to see (q1q2)3=1, and (q1q2q1)2=1. 

6. These HM and DI cusps are examples of how the group theory description of cusps distinguishes them in new ways. For example, in X0(l), the length l cusp is HM and the length 1 cusp is shift of the HM cusp.   

7. The topic of fine moduli: Absolute spaces have fine moduli if G(T,1) is self-normalizing. 

Inner spaces have fine moduli if G has no center. A reduced space component has fine moduli if 

a. Q'' = q1q3-1,sh2 acts as a Klein 4-group on the braid orbit O corresponding to the component. 

b. Neither γ0  nor γ1 has fixed points.  

Example: q1q3-1 acts trivially on an HM rep., and so orbit of Q'' on Ni0+ have length 2. Similarly DI elements in length 1 cusp orbits of Ni0- are fixed by sh2q1q3-1. 


8. This sequence is the abelianization of the Universe l-Frattini cover of G. 

9. The notation here is for the characteristic (abelianized) 2-Frattini quotients of A5, but the same works for the characteristic (abelianized) l-Frattini quotients of G. The centerless and l-perfect condition implies each characteristic quotient is also centerless and l-perfect. 

 10. Use the same notation C for the classes lifted by Schur-Zassenhaus. Among the distinctions between A4 and A5, there is only one conjugacy class of 3-cycles in A5.

11. In the present version, this is [Fr18, Lem. 3.17]. The point of it is that there is a procedure for computing the characteristic l-Frattini modules, and the Schur multiplier (regarded as extremely difficult to find), H2(G,Z), is a detectable quotient of these characteristic modules. 

12. Do the argument that HM reps. have lift invariant 1: ∏(g^1,(g^1)-1, g^3,(g^3)-1). 

13. Full Fried-Serre formula is for any An, and any classes C of odd-order elements (genus 0 family of covers). The proof does reverts to when they are all 3-cycles, for which the result is (-1)r.  A stronger result is that when they are all 3-cycles and r ≥ n, then Ni(An, C3r ) always has two braid orbits separated by the lift invariant. 


14. Reminder: Hk 's are all upper half-plane quotients, but they aren't modular curves, except when G is related to a dihedral group. For r=4, Fried uses a classification of cusps and the genus formula to show how the genus goes up with the tower levels. Note: Nothing to prove unless there is a uniform bound on the definition field of MT  levels. For a given G there are infinitely many MTs with Q as a uniform definition field bound on MT levels. 


15. For simplicity of notation in the OIT, I take r = 4. The target of MTs for higher r is Jr = Ur/PSL2(C). If you can take k0 = 0, then just say it is l-Frattini. 

16. CM stands for complex multiplication, GL2 for open subgroup of  GL2(Zl). It is the interplay of inner and absolute classes on two Nielsen classes that gives a clear picture of the bifurcation between the two types of decomposition groups,  CM and GL2: Ni(Dl,C24),rd and Ni((Z/l)2×xs Z/2,C24),rd.   The braid action on the second Nielsen class is easily computed to be SL2(Zl)/<±1>. Most well-known part of Serre's result – though not said like this – is that SL2(Zl) is l-Frattini for all l > 3. It is eventually l-Frattini for all l. 

17. Because of the use of Falting's Theorem, nothing was explicit about the OIT after its use. Both Fried and Cadaret-Tamagawa use Faltings, too. At this time there is no escaping something replacing it, like a Lang Conjecture, but in this context we are dealing with refined moduli spaces, and canonical polarizations. 


18. There is a serious difference between Q8 and the small Heisenberg group for l=2. In the latter all elements have order 2, while in the former only the center has order 2. Of course, I'm not a group theorist, so I stay close to easier situations, though even there one must have some intuition and computational skill. 

19. Of course, we expect a statement that the first OIT statement has the conclusion l-Frattini for the whole geometric monodromy of the MT, for almost all l. Also, we don't expect the complex multiplication points j' in my system to have the same meaning. Further, the geometry of non-integral j' is already different than in Serre's case.