Zilber–Pink in A2 (special-curve intersections on a Hodge-generic curve)

Prove that for any irreducible curve C in the Siegel modular threefold A2 defined over the algebraic numbers and not contained in a proper special subvariety, the set of points of C lying on special curves inside A2 is finite; formally, establish the finiteness of the union over all special curves Y⊂A2 with dim Y=1 of the intersections C∩Y.

Background

The moduli space A2 parameterizes principally polarized abelian surfaces. In this setting, the special one-dimensional subvarieties (special curves) are of three types: E×CM-curves, E2-curves, and QM-curves. The Zilber–Pink conjecture predicts finiteness of atypical intersections of a Hodge-generic curve with these special loci.

The authors note that positive results toward this conjecture in A2 are currently known only under additional boundary assumptions (via work of Daw and Orr). Their paper develops relations among values of G-functions at quaternionic multiplication (QM) points and derives height bounds that yield partial consequences toward Zilber–Pink, but the full conjecture stated remains unresolved in general.