Zilber–Pink conjecture for curves in abelian varieties

The conjecture concerns finiteness of unlikely intersections between a curve and sufficiently high-codimension algebraic subgroups of an abelian variety. The unrestricted statement is not solved; the newest work claims an explicit reduction bound and a new finiteness argument under additional hypotheses.

Known results

• Habegger--Pila (2014): for curves over a number field not contained in a proper algebraic subgroup, $V(K)\cap X^{[2]}$ is finite.
• Barroero--Gao (2019): reduced the characteristic-$0$ problem to abelian varieties over the algebraic numbers and proved several special cases.
• Barroero--Gao (2019): established the curve statement $V\cap A[2]$ finite unless $V$ lies in a proper algebraic subgroup.
• Dill--Oort (2023): proved a special $A_2$ case for Hodge-generic curves with multiplicative degeneration.

New quantitative bounds

A recent Buium--Coleman preprint claims an explicit reduction bound for Zilber--Pink loci of curves, together with a new finiteness proof and further cardinality bounds. These conclusions are conditional on the paper's stated reduction and abelian-variety hypotheses and do not settle the full conjecture.

Current status (as of August 2026): The number-field curve case and several special characteristic-$0$ cases are established, while the general curve statement remains open; the newest quantitative claims are unverified.