Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture
Abstract Let $${\mathscr {X}} \rightarrow C$$ X → C be a...
| Main Authors: | , , |
|---|---|
| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
Springer Berlin Heidelberg
2022
|
| Online Access: | https://hdl.handle.net/1721.1/142457 |
| _version_ | 1826211541155840000 |
|---|---|
| author | Maulik, Davesh Shankar, Ananth N. Tang, Yunqing |
| author2 | Massachusetts Institute of Technology. Department of Mathematics |
| author_facet | Massachusetts Institute of Technology. Department of Mathematics Maulik, Davesh Shankar, Ananth N. Tang, Yunqing |
| author_sort | Maulik, Davesh |
| collection | MIT |
| description | Abstract
Let
$${\mathscr {X}} \rightarrow C$$
X
→
C
be a non-isotrivial and generically ordinary family of K3 surfaces over a proper curve C in characteristic
$$p \ge 5$$
p
≥
5
. We prove that the geometric Picard rank jumps at infinitely many closed points of C. More generally, suppose that we are given the canonical model of a Shimura variety
$${\mathcal {S}}$$
S
of orthogonal type, associated to a lattice of signature (b, 2) that is self-dual at p. We prove that any generically ordinary proper curve C in
$${\mathcal {S}}_{{\overline{{\mathbb {F}}}}_p}$$
S
F
¯
p
intersects special divisors of
$${\mathcal {S}}_{{\overline{{\mathbb {F}}}}_p}$$
S
F
¯
p
at infinitely many points. As an application, we prove the ordinary Hecke orbit conjecture of Chai–Oort in this setting; that is, we show that ordinary points in
$${\mathcal {S}}_{{\overline{{\mathbb {F}}}}_p}$$
S
F
¯
p
have Zariski-dense Hecke orbits. We also deduce the ordinary Hecke orbit conjecture for certain families of unitary Shimura varieties. |
| first_indexed | 2024-09-23T15:07:38Z |
| format | Article |
| id | mit-1721.1/142457 |
| institution | Massachusetts Institute of Technology |
| language | English |
| last_indexed | 2024-09-23T15:07:38Z |
| publishDate | 2022 |
| publisher | Springer Berlin Heidelberg |
| record_format | dspace |
| spelling | mit-1721.1/1424572023-12-20T16:39:27Z Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture Maulik, Davesh Shankar, Ananth N. Tang, Yunqing Massachusetts Institute of Technology. Department of Mathematics Abstract Let $${\mathscr {X}} \rightarrow C$$ X → C be a non-isotrivial and generically ordinary family of K3 surfaces over a proper curve C in characteristic $$p \ge 5$$ p ≥ 5 . We prove that the geometric Picard rank jumps at infinitely many closed points of C. More generally, suppose that we are given the canonical model of a Shimura variety $${\mathcal {S}}$$ S of orthogonal type, associated to a lattice of signature (b, 2) that is self-dual at p. We prove that any generically ordinary proper curve C in $${\mathcal {S}}_{{\overline{{\mathbb {F}}}}_p}$$ S F ¯ p intersects special divisors of $${\mathcal {S}}_{{\overline{{\mathbb {F}}}}_p}$$ S F ¯ p at infinitely many points. As an application, we prove the ordinary Hecke orbit conjecture of Chai–Oort in this setting; that is, we show that ordinary points in $${\mathcal {S}}_{{\overline{{\mathbb {F}}}}_p}$$ S F ¯ p have Zariski-dense Hecke orbits. We also deduce the ordinary Hecke orbit conjecture for certain families of unitary Shimura varieties. 2022-05-11T12:42:35Z 2022-05-11T12:42:35Z 2022-02-11 2022-05-11T03:28:36Z Article http://purl.org/eprint/type/JournalArticle https://hdl.handle.net/1721.1/142457 Maulik, Davesh, Shankar, Ananth N. and Tang, Yunqing. 2022. "Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture." en https://doi.org/10.1007/s00222-022-01097-x Attribution-NonCommercial-ShareAlike 4.0 International https://creativecommons.org/licenses/by-nc-sa/4.0/ The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature application/pdf Springer Berlin Heidelberg Springer Berlin Heidelberg |
| spellingShingle | Maulik, Davesh Shankar, Ananth N. Tang, Yunqing Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture |
| title | Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture |
| title_full | Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture |
| title_fullStr | Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture |
| title_full_unstemmed | Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture |
| title_short | Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture |
| title_sort | picard ranks of k3 surfaces over function fields and the hecke orbit conjecture |
| url | https://hdl.handle.net/1721.1/142457 |
| work_keys_str_mv | AT maulikdavesh picardranksofk3surfacesoverfunctionfieldsandtheheckeorbitconjecture AT shankarananthn picardranksofk3surfacesoverfunctionfieldsandtheheckeorbitconjecture AT tangyunqing picardranksofk3surfacesoverfunctionfieldsandtheheckeorbitconjecture |