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The positive-characteristic dynamical Mordell–Lang conjecture

Let XX be a quasiprojective variety defined over a field KK of characteristic pp, endowed with an endomorphism Φ\Phi. For a point α∈X(K)\alpha\in X(K) and a subvariety Y⊆XY\subseteq X, define

S:={n∈N0 ⁣:Φn(α)∈Y(K)}.S:=\left\{n\in\mathbb{N}_0\colon \Phi^n(\alpha)\in Y(K)\right\}.

Dynamical Mordell–Lang conjecture. The set SS is a finite union of arithmetic progressions together with finitely many sets of the form

{∑i=1rdipkini ⁣:ni∈N0},\left\{\sum_{i=1}^r d_i p^{k_i n_i}\colon n_i\in\mathbb{N}_0\right\},

for some r∈Nr\in\mathbb{N}, rational numbers did_i, and nonnegative integers kik_i for i=1,…,ri=1,\dots,r.

This is a positive-characteristic variant of the dynamical Mordell–Lang conjecture, motivated by the classical characteristic-zero formulation and by work on dynamics in characteristic pp. The supplied text does not state whether this formulation is known or remains open.

Equivalent formulations 1

Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.

  1. The positive-characteristic dynamical Mordell–Lang conjecture

    Let XX be a variety over an algebraically closed field KK of characteristic p>0p>0, let ff be a rational self-map of XX, let x∈X(K)x\in X(K) be a closed point whose orbit Of(x)\mathcal{O}_{f}(x) is well-defined, and let V⊆XV\subseteq X be a closed subvariety. Positive-characteristic dynamical Mordell–Lang conjecture. The return set

    {n∈N∣fn(x)∈V(K)}\{n\in\mathbb{N}\mid f^{n}(x)\in V(K)\}

    is a pp-normal set in N\mathbb{N}. This was proposed as a positive-characteristic replacement for the complex dynamical Mordell–Lang statement because the latter fails in positive characteristic. The paper disproves this original formulation, so the conjecture is refuted.

    source: Junyi Xie and She Yang, “On the dynamical Mordell-Lang conjecture in positive characteristic”, arXiv:2403.09181 (2024).

Sources & referencesView supporting material

Primary source

Jason Bell and Dragos Ghioca, “A fusion variant of the classical and dynamical Mordell-Lang conjectures in positive characteristic”, arXiv:2205.02644 (2022).

arithmetic-dynamicsdynamical-mordell-langpositive-characteristic

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