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7. Dynatomic side

Let K be a number field f(z)\in K(z) \deg f\geq 2. Let K_f=K(\text{all preperiodic points of }f).


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      Problem 7.1.

      [Nguyen] If K_fK_g is finite over both K_f and K_g, are f and g conjugate, an iterate, or conjugate to something that commutes?


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          Problem 7.2.

          What do you need to know about K_f to figure out what f was (or how close to f can you get)?


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              Problem 7.3.

              [Silverman] Begin to classify those f for which \text{Gal}(K_f/K) does not have finite index in the thing it’s supposed to. Ask this question again for K_f'=K(\text{all periodic points of }f)

                  Cite this as: AimPL: The Galois theory of orbits in arithmetic dynamics, available at http://aimpl.org/galarithdyn.