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## 1. Height functions

Problems related to height functions
1. ### Moduli height

#### 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

#### Problem 1.2.

[Joseph Silverman] Let $f:\mathbb{P}^1 \to \mathbb{P}^1$, not LatteĢs. 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.