Orbifold Mordell conjecture for C-pairs on curves

Prove that for a C-pair (X,Δ) over a number field K with X a smooth proper curve of genus g and 2g − 2 + deg Δ > 0, the C-pair (X,Δ) does not satisfy potential density of integral points.

Background

The conjecture generalizes Mordell–Faltings by incorporating orbifold structures (C-pairs) and predicts arithmetic hyperbolicity in cases of positive orbifold Euler characteristic.

It is known in several related settings: implied by the abc conjecture over number fields, true for higher genus or purely logarithmic divisors via Faltings, and established over function fields.

References

Conjecture [Orbifold Mordell]\label{conj:orb_mor} Let $(X,\Delta)$ be a C-pair over a number field $K$ with $X$ a smooth proper curve over $K$ of genus $g$. Assume that $2g-2 +\deg \Delta >0$. Then the C-pair $(X,\Delta)$ does not satisfy potential density of integral points.

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