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Unlikely intersections with Hecke translates of a special subvariety
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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
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Official URL: https://doi.org/10.4171/JEMS/1005
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 | |||||||||
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| Subjects: | Q Science > QA Mathematics | |||||||||
| Divisions: | Faculty of 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: |
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| Date of first compliant deposit: | 31 July 2018 | |||||||||
| Volume: | 23 | |||||||||
| Number: | 1 | |||||||||
| Page Range: | pp. 1-28 | |||||||||
| DOI: | 10.4171/JEMS/1005 | |||||||||
| Status: | Peer Reviewed | |||||||||
| Publication Status: | Published | |||||||||
| Publisher Statement: | 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 | |||||||||
| RIOXX Funder/Project Grant: |
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