Open Problems in Arithmetic Dynamics — Conjecture 6

Definitions and mathematical statement

Let $K$ be a number field, and fix an algebraic closure $\bar K$. Let $X$ and $S$ be varieties over $K$, let $\pi:X\to S$ be a morphism over $K$, and let $f:X\to X$ be an $S$-morphism, meaning that $\pi\circ f=\pi$. For a finite extension $L/K$ contained in $\bar K$ and a point $s\in S(L)$, write \[ X_s=X\times_{S,s}\operatorname{Spec}L \] for the fiber over $L$, and let $f_s:X_s\to X_s$ be the induced morphism. Its base change to $\bar K$ is the geometric fiber map appearing in the source.

Set $f_s^0=\operatorname{id}_{X_s}$. The set of $L$-rational preperiodic points is \[ \operatorname{PrePer}(f_s,X_s(L)) =\left\{x\in X_s(L):\text{ there exist integers }n\geq 0,\ r\geq 1 \text{ with }f_s^{n+r}(x)=f_s^n(x)\right\}. \] Equivalently, these are the points whose forward orbit under $f_s$ is finite.

Supplied conjectural assertion (Poonen, 2012, Question 3.2; source survey, Conjecture 6). For every choice of $K,X,S,\pi,f$ as above and every integer $D\geq 1$, there exists an integer \[ B=B(K,X,S,\pi,f,D)\geq 0 \] such that, for every finite extension $L/K$ with $[L:K]=D$ and every $s\in S(L)$, \[ \#\operatorname{PrePer}(f_s,X_s(L))\leq B. \] Thus the bound may depend on the fixed family and on $D$, but is independent of $L$ and $s$.

Unresolved qualification. This is the literal assertion supplied in the record. No finiteness hypothesis on the individual preperiodic sets, degree condition on the maps, or restriction excluding identity maps is supplied. The stated hypotheses permit identity maps with infinitely many rational preperiodic points, so the unrestricted formulation cannot be treated as a viable uniform finiteness conjecture without clarification. The intended missing restriction or alternative target is unspecified; none has been inserted into this draft.

Short English statement

For a fixed algebraic family of maps over a number field and a fixed extension degree $D$, the supplied assertion asks for one finite bound on the number of rational preperiodic points in every fiber over every extension of degree exactly $D$. The hypotheses as supplied allow infinite preperiodic sets, so the intended scope requires clarification. The registry labels the problem open; this is supplied metadata, not an independently verified status assessment.

Sources

• UnsolvedMath local registry, permanent ID $800383$, exact source problem code catalogue entry; supplied title: Open Problems in Arithmetic Dynamics --- Conjecture 6. The supplied record has status open and attributes the source list to Benedetto, DeMarco, Ingram, Jones et al., with proposed year 2018.
• Benedetto, DeMarco, Ingram, Jones et al., Current Trends and Open Problems in Arithmetic Dynamics (2018), Conjecture 6. Supplied source URL: https://arxiv.org/abs/1806.04980. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not a claim of independent access or verification.
• The supplied statement attributes the question to Poonen (2012), arXiv:1206.7104, Question 3.2. This reference is preserved as supplied; its original wording and hypotheses have not been independently checked.