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4. Galois Problems

Problems related to Galois Theory
    1. Ramfication in pre-image 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 pull-backs 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 $\mathbb{P}^1$).
      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 $\hat{h}_{crit}$?
      2. 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.

              1. What can you say about inertia subgroups of Galois groups in towers of fields generated by $f^{-n}(\alpha)$ for $f$ PCF?
              2. What happens to decomposition groups?

                  Cite this as: AimPL: Postcritically finite maps in complex and arithmetic dynamics, available at http://aimpl.org/finitedynamics.