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1. Height functions

Problems related to height functions

    1. Moduli height


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      Problem 1.1.

      Let X be a quasi-projective variety and G a group action on X. Let Y=X//G be a geometric quotient of X by G. Is h_Y(x) \asymp \min_{g \in G} h_X(gx)? (ample heights)

      In particular, fix an embedding M_d \subset \mathbb{P}^M. Then define a height h_M:M_d \to \R using the embedding. Is this height function comparable to the minimum height of the maps in the class in \mathop{Rat}_d?

        • Critical height conjecture


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          Problem 1.2.

          [Joseph Silverman] Let f:\mathbb{P}^1 \to \mathbb{P}^1, not Lattès. Define the critical height \begin{equation*} \hat{h}_{crit}(f) = \sum_{c \in \text{crit}(f)} \hat{h}(c). \end{equation*}
          Does there exist c_1,c_2 > 0 depending on d,h_M such that for all f \in M_d(\bar{\mathbb{Q}}) - \{\text{Latt\`es}\} such that \begin{equation*} c_1h_M(f) - c_2 \leq \hat{h}_{crit}(f)? \end{equation*}
              Known for Polynomial maps (Ingram).

          A similar upper bound is known.

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