Skip to content Skip to navigation
University of Warwick
  • Study
  • |
  • Research
  • |
  • Business
  • |
  • Alumni
  • |
  • News
  • |
  • About

University of Warwick
Publications service & WRAP

Highlight your research

  • WRAP
    • Home
    • Search WRAP
    • Browse by Warwick Author
    • Browse WRAP by Year
    • Browse WRAP by Subject
    • Browse WRAP by Department
    • Browse WRAP by Funder
    • Browse Theses by Department
  • Publications Service
    • Home
    • Search Publications Service
    • Browse by Warwick Author
    • Browse Publications service by Year
    • Browse Publications service by Subject
    • Browse Publications service by Department
    • Browse Publications service by Funder
  • Help & Advice
University of Warwick

The Library

  • Login
  • Admin

Unlikely intersections with Hecke translates of a special subvariety

Tools
- Tools
+ Tools

Orr, Martin (2021) Unlikely intersections with Hecke translates of a special subvariety. Journal of the European Mathematical Society, 23 (1). pp. 1-28. doi:10.4171/JEMS/1005 ISSN 1435-9855.

[img]
Preview
PDF
WRAP-unlikely-intersections-Hecke-translates-special-subvariety-Orr-2018.pdf - Accepted Version - Requires a PDF viewer.

Download (845Kb) | Preview
Official URL: https://doi.org/10.4171/JEMS/1005

Request Changes to record.

Abstract

We prove some cases of the Zilber–Pink conjecture on unlikely intersections in Shimura varieties. Firstly, we prove that the Zilber–Pink conjecture holds for intersections between a curve and the union of the Hecke translates of a fixed special subvariety, conditional on arithmetic conjectures. Secondly, we prove the conjecture unconditionally for intersections between a curve and the union of Hecke correspondences on the moduli space of principally polarised abelian varieties, subject to some technical hypotheses. This generalises results of Habegger and Pila on the Zilber–Pink conjecture for products of modular curves.

The conditional proof uses the Pila–Zannier method, relying on a point-counting theorem of Habegger and Pila and a functional transcendence result of Gao. The unconditional results are deduced from this using a variety of arithmetic ingredients: the Masser–Wüstholz isogeny theorem, comparison between Faltings andWeil heights, a super-approximation theorem of Salehi Golsefidy, and a result on expansion and gonality due to Ellenberg, Hall and Kowalski.
The conditional proof uses the Pila-Zannier method, relying on a point-counting theorem of Habegger and Pila and a functional transcendence result of Gao. The unconditional results are deduced from this using a variety of arithmetic ingredients: the Masser-Wüstholz isogeny theorem, comparison between Faltings and Weil heights, a super-approximation theorem of Salehi Golsefidy, and a result on expansion and gonality due to Ellenberg, Hall and Kowalski.

Item Type: Journal Article
Subjects: Q Science > QA Mathematics
Divisions: Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH): Intersection theory (Mathematics), Shimura varieties, Arithmetical algebraic geometry, Hecke algebras
Journal or Publication Title: Journal of the European Mathematical Society
Publisher: European Mathematical Society Publishing House
ISSN: 1435-9855
Official Date: January 2021
Dates:
DateEvent
January 2021Published
5 October 2020Available
6 July 2018Accepted
Volume: 23
Number: 1
Page Range: pp. 1-28
DOI: 10.4171/JEMS/1005
Status: Peer Reviewed
Publication Status: Published
Reuse Statement (publisher, data, author rights): First published in Orr, Martin (2021) Unlikely intersections with Hecke translates of a special subvariety. Journal of the European Mathematical Society . doi:10.4171/JEMS/1005
Access rights to Published version: Restricted or Subscription Access
Copyright Holders: © 2020 EMS Publishing House
Date of first compliant deposit: 31 July 2018
Date of first compliant Open Access: 2 August 2018
RIOXX Funder/Project Grant:
Project/Grant IDRIOXX Funder NameFunder ID
307364FP7 Ideas: European Research Councilhttp://dx.doi.org/10.13039/100011199
EP/M020266/1[EPSRC] Engineering and Physical Sciences Research Councilhttp://dx.doi.org/10.13039/501100000266

Request changes or add full text files to a record

Repository staff actions (login required)

View Item View Item

Downloads

Downloads per month over past year

View more statistics

twitter

Email us: wrap@warwick.ac.uk
Contact Details
About Us