Lang–Vojta and Shafarevich Finiteness Conjectures

Definitions and mathematical statement

Let $K$ be a number field and let $S$ be a finite set of places of $K$ containing all archimedean places. Write \[ \mathcal O_{K,S}=\{a\in K:v(a)\geq 0\text{ for every finite place }v\notin S\}. \] For a $K$-variety $X$, an integral model will mean a separated flat scheme $\mathcal X$ of finite type over $\mathcal O_{K,S}$, equipped with an identification $\mathcal X_K\simeq X$. Its $S$-integral points are the sections \[ \mathcal X(\mathcal O_{K,S})= \operatorname{Hom}_{\mathcal O_{K,S}} \left(\operatorname{Spec}\mathcal O_{K,S},\mathcal X\right). \] These conventions make the dependence of integral points on $S$ and the model explicit; those data are absent from the extracted question.

Integral-point finiteness question. The supplied passage attributes to the Lang--Vojta conjecture the following expectation: for every number field $K$, every finite $S$ as above, every hyperbolic $K$-variety $X$, and every integral model $\mathcal X$ as above, \[ \#\mathcal X(\mathcal O_{K,S})<\infty. \] The word "hyperbolic" is not defined in the record. Thus this is a conditional formulation template: a precise intended conjecture requires specification of the hyperbolicity condition and the allowed varieties. No particular geometric or analytic meaning of hyperbolicity is selected here, and no equivalence with another formulation of Lang--Vojta is asserted.

Shafarevich-type finiteness question. For an explicit smooth projective interpretation, call a smooth projective geometrically connected $K$-variety $V$ canonically polarized when its canonical line bundle $\omega_{V/K}$ is ample. Say that $V$ has good reduction outside $S$ if there exists a smooth proper $\mathcal O_{K,S}$-scheme $\mathcal V$ with generic fibre isomorphic to $V$. These are stated conventions for the template; the record does not specify whether singular varieties, projective integral models, or an extended polarization are intended.

Given a specified class $\mathscr C$ of such varieties, define \[ \mathscr F_{\mathscr C}(K,S)= \left\{[V]_K:\begin{array}{l} V\in\mathscr C\text{ is defined over }K,\\ V\text{ has good reduction outside }S \end{array}\right\}, \] where $[V]_K$ denotes the isomorphism class of the underlying $K$-variety. The classification question in the supplied cleaned statement is: determine classes $\mathscr C$ for which \[ \forall K\;\forall S\supseteq S_\infty(K),\quad S\text{ finite}\ \Longrightarrow\ \#\mathscr F_{\mathscr C}(K,S)<\infty. \] The record supplies neither a particular class nor fixed dimension, Hilbert polynomial, or other numerical invariants. Accordingly, it does not specify a single universal finiteness assertion for all canonically polarized varieties. Nor does it establish a logical implication between this question and the integral-point question.

The original excerpt describes Shafarevich finiteness as known only in very few cases beyond curves and abelian varieties, mentioning K3 surfaces, del Pezzo surfaces, certain hyperkähler varieties, and flag varieties. This is a supplied contextual assessment without the hypotheses of the individual results. These examples do not constitute a list of canonically polarized varieties; they must not be silently incorporated into that restriction.

Short English statement

The record asks whether hyperbolic varieties over number fields have finitely many integral points, and which classes of canonically polarized varieties have finitely many isomorphism classes with prescribed good reduction. The displayed templates specify integral models and quantifiers, but the intended hyperbolicity condition, classes, and boundedness hypotheses remain unresolved. This is a statement-only draft requiring human mathematical review.

Sources

• Supplied local UnsolvedMath registry record: permanent ID $30006079$, exact source problem code catalogue entry, title "Lang--Vojta and Shafarevich Finiteness Conjectures," Oberwolfach Reports Open Problems collection. Project registry locations are data/unsolvedmath/problems.json and catalog/problems.json; this draft uses the supplied complete record without inspecting those files.
• Supplied source citation: Anabelian Geometry and Representations of Fundamental Groups (2025), Oberwolfach Reports, DOI 10.4171/owr/2024/45, pp. 2621--2670. The extract carries page number 2661. Source and literature link: https://doi.org/10.4171/owr/2024/45. The record separately gives proposed year 2024 and report designation 45/2024; these are retained without resolving their relationship to the citation year 2025.
• Attribution context: the extract includes the heading "Arithmetic Finiteness and Big Monodromy on Abelian Varieties," Thomas Krämer, joint work with Ariyan Javanpeykar, Christian Lehn, and Marco Maculan. The record names no proposer, and the placement of this heading alone does not establish authorship of the preceding passage or of the conjectures.
• Supplied additional literature source: Lawrence--Sawin, "Shafarevich for hypersurfaces," https://annals.math.princeton.edu/2025/202-3/p01. No precise theorem or hypotheses from this reference are supplied.
• Literature assessment dated exactly 2026-08-22: the supplied curation labels the problem open, describes Lang--Vojta finiteness as unresolved, and describes Shafarevich results as restricted and recent results as partial. This is supplied dated curation, not an independently verified literature assessment. The background explicitly requests independent verification of time-sensitive status and cited claims.
• The registry labels the statement corrected_verified and says that the official report was checked and the question isolated with repaired notation. These are supplied source-verification claims; no source visit or independent verification is asserted by this draft.