
## 2. Arithmetic Questions

1. #### Problem 2.05.

[P. Ingram, H. Krieger] Do “rigid” – analogues of maps that are not flexible Lattès – PCF maps form a subset of bounded height in $\mathcal{M}_d^N(\overline{\Bbb{Q}})$?
1. Remark. Known for $N=1$ [MR3265554].
• #### Problem 2.1.

[L. DeMarco and J. Silverman] Is there a countable Zariski dense (Galois invariant) subset of “dynamically interesting” maps in $\mathcal{M}_d^N(\overline{\Bbb{Q}})$ (with bounded height)?
• #### Problem 2.15.

[H. Krieger] What is the locus in $\mathcal{M}_d^N$ of maps with “small” arboreal image? More precisely, what can be said about the locus in $\mathcal{M}_d^N\times\Bbb{P}^N$ formed by $(f,P)$ such that the image of the associated arboreal representation is of infinite index? Do such points lie on any “dynamically special” subvariety?
1. Remark. When $N=1$, we know that the image of the arboreal representation is small for PCF maps or for maps with non-trivial automorphisms.
• #### Problem 2.2.

[J. Silverman] For a map $f$ defined over a field $K$, let $K^{\rm{pre}}_f$ be the extension of $K$ generated by all preperiodic points of $f$. What can be said about $[K^{\rm{pre}}_f:K]$? What $K^{\rm{pre}}_f=K^{\rm{pre}}_g$ implies about maps $f$ and $g$? What if $K^{\rm{pre}}$ is replaced with $K^{\rm{preper}}$ or $K^{\rm{mult}}$, extensions generated by preperiodic points, or multipliers of periodic points?
• #### Problem 2.25.

[H. Krieger, J. Silverman] The Shafarevich dimension of $\mathcal{M}_d^N$ is defined as $$\sup_{(K,S)}\dim \overline{\left\{\text{elements of } \mathcal{M}_d^N(K) \text{ that have good reduction outside } S\right\}}.$$ What is this dimension? Is it equal to $2d-2$ when $N=1$?
1. Remark. The Shafarevich dimension of $\mathcal{M}_d^1$ is known to be at least $d+1$ when $d\geq 3$ [MR3778330].
• #### Problem 2.3.

[C. Petsche, B. Thompson] Generalizations of dynamical local/global\\ PCF (post-critically finite) $\Leftrightarrow$ PCB (post-critically bounded)?
• #### Problem 2.35.

[N. Looper] Find an appropriate critical/canonical height on $\mathcal{M}_d^N$.
• #### Problem 2.4.

[N. Looper] Find/prove dynamical analogues of the Szpiro conjecture.
1. Remark. In number theory, Szpiro’s conjecture relates the conductor and the discriminant of an elliptic curve.
• #### Problem 2.45.

[H. Krieger] For $\alpha\in\bar{\Bbb{Q}}$, investigate $\max\{\hat{\lambda}_{\nu,f}(\sigma\alpha)\mid \sigma\in{\rm{Gal}}(\bar{\Bbb{Q}}/\Bbb{Q})\}$ ($\hat{\lambda}_{\nu,f}$ the local height associated with a map $f$ and a valuation $\nu$).
• #### Problem 2.5.

[A. Shankar] Lift ${\rm{End}}^N_d(\overline{\Bbb{F}_p})$ to PCF elements of ${\rm{End}}^N_d(\overline{\Bbb{Q}_p})$.
1. Remark. This is in analogy with Deuring’s theorem on elliptic curves.
• #### Problem 2.55.

[J. Silverman] Is there a bound for $[K_{\rm{def}}:K_{\rm{moduli}}]$ depending on $N$ but not on $d$?
1. Remark. When $N=1$, the bound is $2$ [MR3230378]
• Remark. There exists a uniform bound depending on both $N$ and $d$ [MR3867431].
• #### Problem 2.6.

[J. Silverman] Study dynamical systems $\Bbb{P}^N\rightarrow\Bbb{P}^N$ whose field of moduli is contained in $\Bbb{R}$. When can they be defined over the reals?
1. Remark. The case of $N=1$ is thoroughly studied [MR4295550].

Cite this as: AimPL: Moduli spaces for algebraic dynamical systems, available at http://aimpl.org/modalgdyn.