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 -