We prove the abelian Zilber–Pink conjecture over $\overline{\mathbb Q}$. Every irreducible subvariety of an abelian variety has only finitely many maximal atypical subvarieties, where atypicality is measured inside its smallest containing torsion coset.

We prove the $E\times\mathrm{CM}$ component of Zilber–Pink for Hodge-generic algebraic curves in $\mathcal A_2$ over $\overline{\mathbb Q}$. Each such curve contains only finitely many points whose abelian surface is isogenous to a product of elliptic curves with at least one factor having complex multiplication.

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.

We prove that every Hodge-generic algebraic curve in $\mathcal A_2$ over $\overline{\mathbb Q}$ contains only finitely many points whose abelian surface is isogenous to the square of an elliptic curve without complex multiplication (CM). Combining this result with the companion $E\times\mathrm{CM}$ and quaternionic-division finiteness theorems, we prove the curve case of Zilber–Pink in $\mathcal A_2$ over $\overline{\mathbb Q}$, without a boundary hypothesis.