Colloquium: Connected components of Sphere covers

Mike Fried, Emeritus Professor UC Irvine


By 1872 we knew the space of compact surfaces of genus g was connected. I discuss applications extending this. Consider pairs (Xi,fi) with fi, nonconstant and analytic, mapping Xi to the sphere, i=1,2, both with r > 3 branch points. 


A locally path connected topology on such pairs uses dragging the branch points of the fi s. We  ask: 

When is (X1,f1)  connected to (X2,f2)? 

We find the fi s must have the same geometric monodromy group G with the same r conjugacy classes attached to the branch points. Sometimes these invariants suffice; we get one connected component.  Sometimes we need more sophisticated invariants to describe components. 


Two problems on rational functions f with coefficients, in a number field K, allow me to show the tools that make this work.  


1. Find all (f,K) with the Schur cover property: For infinitely many residue fields of K, f : w → f(w)=z, with w running over the residue field (including ∞), is one-one. 


2. Find all (f,K), with f indecomposable over K, but f is a composition of degree > 1 rational functions over the complexes. 


We find G is one of two "easy" (centerless) groups: The dihedral group of order 2n (n odd) or another slightly larger group, and all r = 4 conjugacy classes are the same. 

  

Then we see that modular curves include all rational functions giving solutions. Finally, the solutions come from the two fiber types over the j-line in Serre's Open Image Theorem. 

This talk is based on §6.1 and §6.2 of "The place of exceptional covers among all diophantine relations," J. Finite Fields 11 (2005) 367–433, arXiv:0910.3331v1 [math.NT]  http://www.math.uci.edu/~mfried/paplist-ff/exceptTower0910-3331v1.pdf


These are footnote comments, per numbers listed in the pdf file, that appear during the talk.


0. Geometry Talk 1 (2nd talk) does a special case of the construction of spaces connected to Riemann's deepest contribution to complex variables, and the space of curves of genus g. Geometry Talk 2 turns it around and produces an example that is a true generalizing situation for the OIT.

1. Give a picture of dragging the branch points: at the board. A rephrasing of part of Riemann's Existence Theorem. 


2. Inner equivalence will appear in a big way in the 1st Geometry talk: In all my examples, these abs and inn spaces extend the most classical pair of modular curves X0, X1. 


3. The cover φg: Wg → P1z is connected equivalent to monodromy group Gg  transitive in natural deg(φg) permutation representation;  Wg  is isomorphic to P1 when it has genus 0 (genus of an element in a Nielsen class).  

4. Tf  is imprimitive if there is a group properly between Gf and Gf (1), the stabilizer of an element in the Tf  action.


5. Reduced equivalence: I went to the board to explain PGL2 action through composition of φg with linear fractional transformations. You want that equivalence to significantly have infinitely many examples of exceptional covers.  


6. Degree p exceptional covers must have geometric monodromy a transitive subgroup of  Z/p xs (Z/p)*, or else it would be double transitive, violating the characterization:  Burnside's Theorem. 

What happens for r=4 when you compute the genus g  of the covers in the family. The minimal index of an element is (p-1)/2. That happens when it is an involution. So they must all be in the involution class:

 

2(p + g -1) = 4(p - 1)/2. That is,  g is 0. 


7.  Start of discussion of geometric monodromy versus arithmetic monodromy as parallel to inner equivalence of covers versus absolute equivalence.  Geometric monodromy is the Galois group of a cover over the algebraic closure. Arithmetic monodromy is that over its field of definition.

7a. Note: Z/p xs (Z/p)*is is the subgroup of Sp that normalizes Dp. So modding out by it on Nielsen classes defines absolute equivalence. 

7b. Geometric monodromy of a cover is a normal subgroup of Arithmetic monodromy. For an exceptional cover, they cannot be equal. A major theme of the Geometry talks is that the solution spaces will distinguish arithmetic and geometric monodromy. This generalizes a theme behind the theory of Complex Multiplication. 

8.  * = abs  or inn: Ni(G,C)in formed by modding out by conjugation action of G on Nielsen classes. In Geometry 1 talk, I explain what is the equivalence on P1z covers: These are Galois covers with group G. 

Ramification over j = 0 divides 3, over j =1 divides 2. Ramification over \infty are the cusps. We  easily compute this. For X0(p) you get a length 1 and length p cusp. 


9. Q'' acts trivially for modular curves. That's why you haven't seen this group before. It is crucial for interpreting properties of the spaces, even for modular curves. For example, the Hurwitz space as a U4 cover has fine moduli. But its reduced version does not, because the first criterion for that is that Q'' acts on Nielsen classes through a Klein 4-group.[BFr02, §4.3]