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The dynamical Mordell–Lang conjecture

About 19 years old · traced to

Let f:X→Xf:X\to X be an endomorphism of a quasi-projective variety over a field KK of characteristic 00, and let VV be a closed subvariety of XX. For x∈X(K)x\in X(K), define the return set

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

Dynamical Mordell–Lang conjecture. For every x∈X(K)x\in X(K), the return set is a finite union of arithmetic progressions. This conjecture is a central problem in arithmetic dynamics. The paper proves it for products of endomorphisms of an affine curve and a projective curve over Q‾\overline{\mathbb{Q}}, while the general statement remains open.

References

Primary source

Junyi Xie, She Yang and Aoyang Zheng, “Dynamical Mordell-Lang conjecture for split self-maps of affine curve times projective curve”, arXiv:2602.08608 (2026).

Additional references

22 papers in this index state this conjecture (2007–2026). The statement above is taken from the most recent of them; the others are arXiv:2409.07826, arXiv:2403.05107, arXiv:2307.05885, arXiv:2302.11736, arXiv:2202.01673, arXiv:2201.12750, arXiv:2107.03559, arXiv:2102.04073, arXiv:2012.13711, arXiv:1802.05309, arXiv:1801.02831, arXiv:1706.06470, and 9 more.

algebraic-geometryarithmetic-dynamicsdynamical-mordell-lang
Posted · edited

Progress summary

Refreshed
Claimed progress

New 2026 papers settle substantial cases of the conjecture, but the general problem remains open.

The conjecture, proposed by Ghioca and Tucker, predicts that each return set is a finite union of arithmetic progressions. The general statement for characteristic-zero quasi-projective varieties is not proved.

Known results

  • Étale dominant morphisms of quasi-projective varieties in characteristic zero.
  • Quadratic-polynomial dynamics, for subvarieties of arbitrary dimension.
  • Various specialized results for skew-linear maps, semiabelian varieties, and positive-characteristic systems.

2026 partial advances

A preprint proves the conjecture for split maps on a product of an affine curve and a projective curve over Q‾\overline{\mathbb{Q}}. Another claims it for relatively étale systems over products of smooth projective curves over C\mathbb{C}. Yang and Zheng additionally establish a criterion for regular endomorphisms of affine space based on multiplicities at infinity. These are substantial special cases, not a general proof, and the claims have not received independent verification in the retrieved sources.

Current status (as of October 2026): substantial classes are claimed solved, but the general dynamical Mordell–Lang conjecture remains open and the recent claims are unverified.

Sources

Solutions 0

No solutions have been posted yet.