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 - 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 -