Unlikely Intersections with ExCM curves in A2

Christopher Daw, Martin Orr

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Abstract

The Zilber-Pink conjecture predicts that an algebraic curve in A2 has only finitely many intersections with the special curves, unless it is contained in a proper special subvariety. Under a large Galois orbits hypothesis, we prove the finiteness of the intersection with the special curves parametrising abelian surfaces isogenous to the product of two elliptic curves, at least one of which has complex multiplication. Furthermore, we show that this large Galois orbits hypothesis holds for curves satisfying a condition on their intersection with the boundary of the Baily-Borel compactification of A2.

More generally, we show that a Hodge generic curve in an arbitrary Shimura variety has only finitely many intersection points with the generic points of a Hecke-facteur family, again under a large Galois orbits hypothesis.
Original languageEnglish
Pages (from-to)1705-1745
Number of pages41
JournalAnnali della Scuola Normale di Pisa - Classe di Scienze
DOIs
Publication statusPublished - 22 Dec 2021

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