On the generalised André–Pink–Zannier conjecture.
We introduce and study the notion of a generalised Hecke orbit in a Shimura variety. We define a height function on such an orbit and study its properties. We obtain lower bounds for the sizes of Galois orbits of points in a generalised Hecke orbit in terms of this height function, assuming the “wea...
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
Académie des sciences
2023-12-01
|
| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.491/ |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1825206248688058368 |
|---|---|
| author | Richard, Rodolphe Yafaev, Andrei |
| author_facet | Richard, Rodolphe Yafaev, Andrei |
| author_sort | Richard, Rodolphe |
| collection | DOAJ |
| description | We introduce and study the notion of a generalised Hecke orbit in a Shimura variety. We define a height function on such an orbit and study its properties. We obtain lower bounds for the sizes of Galois orbits of points in a generalised Hecke orbit in terms of this height function, assuming the “weakly adelic Mumford–Tate hypothesis” and prove the generalised André–Pink–Zannier conjecture (a special case of the Zilber-Pink conjecture) under this assumption using the Pila–Zannier strategy. |
| format | Article |
| id | doaj-art-932991546e9548f297a5eb9e286886ee |
| institution | Kabale University |
| issn | 1778-3569 |
| language | English |
| publishDate | 2023-12-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj-art-932991546e9548f297a5eb9e286886ee2025-02-07T11:12:14ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692023-12-01361G111717172210.5802/crmath.49110.5802/crmath.491On the generalised André–Pink–Zannier conjecture.Richard, Rodolphe0Yafaev, Andrei1UCL Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UKUCL Department of Mathematics, University College London, Gower Street, London, WC1E 6BT, UKWe introduce and study the notion of a generalised Hecke orbit in a Shimura variety. We define a height function on such an orbit and study its properties. We obtain lower bounds for the sizes of Galois orbits of points in a generalised Hecke orbit in terms of this height function, assuming the “weakly adelic Mumford–Tate hypothesis” and prove the generalised André–Pink–Zannier conjecture (a special case of the Zilber-Pink conjecture) under this assumption using the Pila–Zannier strategy.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.491/Shimura varietiesHecke orbitsZilber-PinkHeightsSiegel setsMumford–Tate conjectureAdelic linear groups |
| spellingShingle | Richard, Rodolphe Yafaev, Andrei On the generalised André–Pink–Zannier conjecture. Comptes Rendus. Mathématique Shimura varieties Hecke orbits Zilber-Pink Heights Siegel sets Mumford–Tate conjecture Adelic linear groups |
| title | On the generalised André–Pink–Zannier conjecture. |
| title_full | On the generalised André–Pink–Zannier conjecture. |
| title_fullStr | On the generalised André–Pink–Zannier conjecture. |
| title_full_unstemmed | On the generalised André–Pink–Zannier conjecture. |
| title_short | On the generalised André–Pink–Zannier conjecture. |
| title_sort | on the generalised andre pink zannier conjecture |
| topic | Shimura varieties Hecke orbits Zilber-Pink Heights Siegel sets Mumford–Tate conjecture Adelic linear groups |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.491/ |
| work_keys_str_mv | AT richardrodolphe onthegeneralisedandrepinkzannierconjecture AT yafaevandrei onthegeneralisedandrepinkzannierconjecture |