Quaternionic division points on curves in the Siegel threefold

We prove the quaternionic-division component of Zilber–Pink for curves in ${\mathcal{A}}_2$ over $\overline{\mathbb{Q}}$. A Hodge-generic algebraic curve contains only finitely many points whose full geometric rational endomorphism algebra is an indefinite quaternion division algebra over ℚ. No boundary or reduction hypothesis is required.