4. Galois Problems
Problems related to Galois Theory
Ramfication in preimage towers
Let $X$ be a projective variety, $f:X/K \to X/K$ PCF. Let $\alpha \in X(K)$. Look at the tower of fields from adding the pullbacks of $\alpha$ \begin{equation*} K_n = K(f^{n}(\alpha)). \end{equation*}Conjecture 4.1.
[Rafe Jones] There exists a finite set of primes $S$ such that all $K_n$ are unramified outside of $S$.
(Known for $\mathbb{P}^1$). If not PCF and $X=\mathbb{P}^1$ let $S_n = \{\text{primes of $K$ where $K_n/K$ is ramified}\}$. Is the growth rate of $\#S_n$ related to some height of $f$, specifically $\hat{h}_{crit}$?
 Assume conjecture ($X= \mathbb{P}^1$). Can you arrange it so that all of the ramification in the tower is tame? (even for one example).

Detecting PCF maps
Problem 4.2.
Let $f$ be a PCF rational function defined over a number field. Is it “algebraically detectable” from $K_n = K(\mathop{Per}_n(f))$ or the tower of such extensions that $f$ is PCF. Galois groups? Ramification? 
inertia groups for preimage towers
Problem 4.3.
 What can you say about inertia subgroups of Galois groups in towers of fields generated by $f^{n}(\alpha)$ for $f$ PCF?
 What happens to decomposition groups?
Cite this as: AimPL: Postcritically finite maps in complex and arithmetic dynamics, available at http://aimpl.org/finitedynamics.