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Unlikely intersections with ExCM curves in A2
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Daw, Christopher and Orr, Martin (2021) Unlikely intersections with ExCM curves in A2. Annali della Scuola Normale di Pisa - Classe di Scienze, XXII (4). pp. 1705-1745. doi:10.2422/2036-2145.202006_014 ISSN 0391-173X.
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Official URL: https://doi.org/10.2422/2036-2145.202006_014
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 conjecture, 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 conjecture 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 conjecture.
Item Type: | Journal Article | ||||||||||||
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Subjects: | Q Science > QA Mathematics | ||||||||||||
Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||||||
Library of Congress Subject Headings (LCSH): | Abelian varieties, Shimura varieties , Curves, Algebraic, Arithmetical algebraic geometry | ||||||||||||
Journal or Publication Title: | Annali della Scuola Normale di Pisa - Classe di Scienze | ||||||||||||
Publisher: | Scuola Normale Superiore di Pisa | ||||||||||||
ISSN: | 0391-173X | ||||||||||||
Official Date: | 22 December 2021 | ||||||||||||
Dates: |
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Volume: | XXII | ||||||||||||
Number: | 4 | ||||||||||||
Number of Pages: | 39 | ||||||||||||
Page Range: | pp. 1705-1745 | ||||||||||||
DOI: | 10.2422/2036-2145.202006_014 | ||||||||||||
Status: | Peer Reviewed | ||||||||||||
Publication Status: | Published | ||||||||||||
Access rights to Published version: | Restricted or Subscription Access | ||||||||||||
Date of first compliant deposit: | 25 September 2020 | ||||||||||||
Date of first compliant Open Access: | 25 September 2020 | ||||||||||||
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