1. Height functions
Problems related to height functions-
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
Known for Polynomial maps (Ingram).Problem 1.2.
[Joseph Silverman] Let $f:\mathbb{P}^1 \to \mathbb{P}^1$, not LatteĢs. 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*}
A similar upper bound is known.
Cite this as: AimPL: Postcritically finite maps in complex and arithmetic dynamics, available at http://aimpl.org/finitedynamics.