Weakly Special Conjecture

Determine whether every weakly special variety X over a number field K has a potentially dense set of integral points.

Background

Motivated by the converse of the Lang–Vojta + Chevalley–Weil implication, the authors record a broad conjecture relating weak specialness to potential density of integral points.

Although verified for several classes (e.g., abelian varieties, certain K3 surfaces, Calabi–Yau threefolds), it is expected to fail in general, making its full resolution open.

References

Conjecture [Weakly Special Conjecture]\label{conj:ws} If $X$ is a weakly special variety over a number field $K$, then $X$ has a potentially dense set of integral points.

— Weakly special varieties, Campana stacks, and Remarks on Orbifold Mordell  (2603.28745 - Bartsch et al., 30 Mar 2026) in Introduction, Special varieties (Conjecture \ref{conj:ws})