The Arithmetic Persistence Conjecture: does arithmetic hyperbolicity persist under extension of the base field?

The Arithmetic Persistence Conjecture of Javanpeykar predicts that arithmetic hyperbolicity — finiteness of integral points on every model over every Z-finitely generated subring of the base field — is stable under extension of the algebraically closed base field: if X is an arithmetically hyperbolic finite type separated scheme over an algebraically closed field k of characteristic zero, then X_L is arithmetically hyperbolic over L for every extension of algebraically closed fields k ⊂ L. In concrete Diophantine terms, it asks whether a system of polynomial equations having only finitely many solutions in the S-integers of every number field must have only finitely many solutions in every Z-finitely generated integral domain of characteristic zero. For proper varieties the conjecture is a formal consequence of the Green–Griffiths–Lang/Lang–Vojta conjectures. It is known for several geometric classes — Brody hyperbolic projective varieties, varieties with a finite map to an abelian variety, projective surfaces admitting a non-constant map to an abelian variety, varieties with a quasi-finite map to a semi-abelian variety, hyperbolically embeddable affine varieties, varieties with a quasi-finite period map, moduli of polarized varieties with semi-ample canonical bundle, and (by Morrow) all projective varieties whose integral subvarieties are of general type — but remains open in general.

Background

In Diophantine geometry one studies integer-like solutions of polynomial systems. For a number field $K$ (a finite extension of $\mathbb{Q}$ ) and a finite set $S$ of finite places of $K$ , the ring of $S$ -integers $\mathcal{O}_{K,S}$ consists of the elements of $K$ that are integral at all finite places outside $S$ ; when $S$ is empty it is the usual ring of integers. A recurring phenomenon, going back to Siegel and to the Mordell conjecture for curves of genus at least two, is that many varieties have only finitely many $\mathcal{O}_{K,S}$ -points. Javanpeykar axiomatized this as follows. Let $k$ be an algebraically closed field of characteristic zero. A model for a finite type separated scheme $X$ over $k$ over a subring $A\subset k$ is a finite type separated scheme $\mathcal{X}\to\operatorname{Spec}A$ together with an isomorphism $\mathcal{X}_k\cong X$ ; informally, $\mathcal{X}$ is the same variety cut out by equations whose coefficients all lie in $A$ . One says that $X$ is arithmetically hyperbolic over $k$ if $\mathcal{X}(A)$ is finite for every $\mathbb{Z}$ -finitely generated subring $A\subset k$ (a subring finitely generated as a $\mathbb{Z}$ -algebra) and every model $\mathcal{X}$ of $X$ over $A$ . For an affine variety $X\subset\mathbb{A}^n_{\overline{\mathbb{Q}}}$ defined by polynomial equations, arithmetic hyperbolicity over $\overline{\mathbb{Q}}$ says exactly that the system has only finitely many solutions in $\mathcal{O}_{K,S}^n$ for every number field $K$ and every finite $S$ , while arithmetic hyperbolicity over $\mathbb{C}$ demands finitely many solutions in $A^n$ for every $\mathbb{Z}$ -finitely generated integral domain $A\subset\mathbb{C}$ of characteristic zero, for instance $A=\mathbb{Z}[t_1,\dots,t_r]$ . These conditions are a priori very different: the second quantifies over a vastly larger class of rings obtained by allowing transcendental coefficients, so finiteness over number rings does not obviously persist when the base field grows. Classical results verify the persistence in examples: by the Mordell conjecture and its function-field analogue, smooth projective curves of genus at least two are arithmetically hyperbolic over any algebraically closed field of characteristic zero, and work of Lang on $\mathbb{P}^1\setminus\{0,1,\infty\}$ establishes the analogous statement for this punctured line.

Arithmetic hyperbolicity is the arithmetic member of a web of conjecturally equivalent notions organized by the Green–Griffiths–Lang and Lang–Vojta conjectures. On the complex-analytic side, a variety is Brody hyperbolic if it carries no nonconstant holomorphic map from $\mathbb{C}$ ; on the algebraic side, a projective variety is of general type if its canonical bundle has maximal growth (its spaces of pluricanonical sections grow as fast as dimension allows). The Green–Griffiths–Lang conjecture predicts that a projective variety over $\overline{\mathbb{Q}}$ is arithmetically hyperbolic if and only if every integral subvariety (every positive-dimensional closed subvariety) is of general type. Since being of general type is visibly stable under extending the algebraically closed base field, that conjecture implies, for proper $X$ , that arithmetic hyperbolicity should also be stable under base change. Javanpeykar isolated this consequence as an independent question, the Persistence Conjecture (also called the Arithmetic Persistence Conjecture), stated for arbitrary finite type separated schemes — including affine and otherwise non-proper ones — where no reduction to the general-type mechanism is available. The conjecture is reasonable but resistant: all known proofs require an auxiliary geometric boundedness property of the variety, the weakest of which, mild boundedness, says roughly that after imposing finitely many base points on any smooth curve, the pointed maps from that curve into the variety form a finite set, and it is itself only conjectured to hold for all arithmetically hyperbolic varieties.

Problem Statement

Let $k\subset L$ be an extension of algebraically closed fields of characteristic zero, and let $X$ be a finite type separated scheme over $k$ which is arithmetically hyperbolic over $k$ ; that is, for every $\mathbb{Z}$ -finitely generated subring $A\subset k$ and every model $\mathcal{X}$ of $X$ over $A$ , the set $\mathcal{X}(A)$ is finite. The Arithmetic Persistence Conjecture asserts that $X_L$ is arithmetically hyperbolic over $L$ . Prove this in full generality, or refute it by exhibiting an arithmetically hyperbolic scheme $X$ over some algebraically closed field $k$ of characteristic zero and an extension $L/k$ for which $X_L$ is not arithmetically hyperbolic over $L$ . In concrete Diophantine terms (the affine case over $\overline{\mathbb{Q}}$ , a special case of the abstract formulation above): if a system of polynomial equations over $\overline{\mathbb{Q}}$ has only finitely many solutions in $\mathcal{O}_{K,S}$ for every number field $K$ and every finite set $S$ of finite places, must it have only finitely many solutions in every $\mathbb{Z}$ -finitely generated integral domain of characteristic zero? Both the proper and the non-proper (quasi-projective, affine) cases are in scope, and the statement must be established without auxiliary geometric hypotheses on $X$ — no Brody or algebraic hyperbolicity, no mild boundedness, no assumption that all integral subvarieties are of general type, no morphism to an abelian or semi-abelian variety, no quasi-finite period map — since those restricted cases are precisely what is already known.

References

Ariyan Javanpeykar, 'Arithmetic hyperbolicity: automorphisms and persistence', Mathematische Annalen 381 (2021), no. 1-2, 439-457. DOI: 10.1007/s00208-021-02155-0 (arXiv:1809.06818). https://doi.org/10.1007/s00208-021-02155-0
Ariyan Javanpeykar, 'The Lang-Vojta Conjectures on Projective Pseudo-Hyperbolic Varieties', in 'Arithmetic Geometry of Logarithmic Pairs and Hyperbolicity of Moduli Spaces' (CRM Short Courses), Springer, 2020, pp. 135-196. DOI: 10.1007/978-3-030-49864-1_3 (arXiv:2002.11981). https://doi.org/10.1007/978-3-030-49864-1_3
Raymond van Bommel, Ariyan Javanpeykar, Ljudmila Kamenova, 'Boundedness in families with applications to arithmetic hyperbolicity', Journal of the London Mathematical Society 109 (2024), no. 1, article e12847. DOI: 10.1112/jlms.12847 (arXiv:1907.11225). https://doi.org/10.1112/jlms.12847
Jackson S. Morrow, 'Boundedness of hyperbolic varieties', preprint, 2022. arXiv:2209.09982, DOI: 10.48550/arXiv.2209.09982. https://arxiv.org/abs/2209.09982
Ariyan Javanpeykar, Aaron Levin, 'Urata's theorem in the logarithmic case and applications to integral points', Bulletin of the London Mathematical Society 54 (2022), no. 5, 1772-1790. DOI: 10.1112/blms.12655 (arXiv:2002.11709). https://doi.org/10.1112/blms.12655
Ariyan Javanpeykar, Daniel Litt, 'Integral points on algebraic subvarieties of period domains: from number fields to finitely generated fields', Manuscripta Mathematica 173 (2023), no. 1-2, 23-44. DOI: 10.1007/s00229-023-01463-w (arXiv:1907.13536). https://doi.org/10.1007/s00229-023-01463-w
Ariyan Javanpeykar, Ruiran Sun, Kang Zuo, 'The Shafarevich conjecture revisited: Finiteness of pointed families of polarized varieties', preprint, 2020. arXiv:2005.05933. https://arxiv.org/abs/2005.05933
Ariyan Javanpeykar, 'Finiteness of non-constant maps over number fields', preprint, 2021. arXiv:2112.11408. https://arxiv.org/abs/2112.11408
Arno Fehm, Ariyan Javanpeykar, 'Hilbert properties of varieties', preprint, 2025. arXiv:2511.18431. https://arxiv.org/abs/2511.18431
Finn Bartsch, Ariyan Javanpeykar, Erwan Rousseau, 'The theorem of Maehara-Severi for maps of general type', preprint, 2026. arXiv:2602.02314. https://arxiv.org/abs/2602.02314