TY - JOUR

T1 - Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture

AU - Maulik, Davesh

AU - Shankar, Ananth N.

AU - Tang, Yunqing

N1 - Funding Information:
We thank George Boxer, Ching-Li Chai, Johan de Jong, Kai-Wen Lan, Keerthi Madapusi Pera, Frans Oort, Arul Shankar, Andrew Snowden, Salim Tayou, and Tonghai Yang for helpful discussions, as well as Arthur and D.W. Read for additional assistance. A.S. has been partially supported by the NSF Grant DMS-2100436 and Y.T. has been partially supported by the NSF Grant DMS-1801237. Y.T. was a chargée de recherche at CNRS and Université Paris-Saclay from February 2020 to June 2021. We would like to thank the referee for a careful and thorough reading, and for valuable suggestions which vastly improved the exposition of the paper.
Funding Information:
We thank George Boxer, Ching-Li Chai, Johan de Jong, Kai-Wen Lan, Keerthi Madapusi Pera, Frans Oort, Arul Shankar, Andrew Snowden, Salim Tayou, and Tonghai Yang for helpful discussions, as well as Arthur and D.W. Read for additional assistance. A.S. has been partially supported by the NSF Grant DMS-2100436 and Y.T. has been partially supported by the NSF Grant DMS-1801237. Y.T. was a chargée de recherche at CNRS and Université Paris-Saclay from February 2020 to June 2021. We would like to thank the referee for a careful and thorough reading, and for valuable suggestions which vastly improved the exposition of the paper.
Publisher Copyright:
© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.

PY - 2022/6

Y1 - 2022/6

N2 - Let X→ C be a non-isotrivial and generically ordinary family of K3 surfaces over a proper curve C in characteristic 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 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 SF¯p intersects special divisors of SF¯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 SF¯p have Zariski-dense Hecke orbits. We also deduce the ordinary Hecke orbit conjecture for certain families of unitary Shimura varieties.

AB - Let X→ C be a non-isotrivial and generically ordinary family of K3 surfaces over a proper curve C in characteristic 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 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 SF¯p intersects special divisors of SF¯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 SF¯p have Zariski-dense Hecke orbits. We also deduce the ordinary Hecke orbit conjecture for certain families of unitary Shimura varieties.

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U2 - 10.1007/s00222-022-01097-x

DO - 10.1007/s00222-022-01097-x

M3 - Article

AN - SCOPUS:85124729183

SN - 0020-9910

VL - 228

SP - 1075

EP - 1143

JO - Inventiones Mathematicae

JF - Inventiones Mathematicae

IS - 3

ER -