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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*}

      Add remark

      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


          Add remark

          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


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              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.