Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture
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Author(s)
Maulik, Davesh
Shankar, Ananth N.
Tang, Yunqing
Date Issued
February 11, 2022
Publisher
Springer Berlin Heidelberg
Citation
Maulik, Davesh, Shankar, Ananth N. and Tang, Yunqing. 2022. "Picard ranks of K3 surfaces over function fields and the Hecke orbit conjecture."
Version
Author's final manuscript
Abstract
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.
MIT Department
Massachusetts Institute of Technology. Department of Mathematics
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DOI of Published Version
https://doi.org/10.1007/s00222-022-01097-x
