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

Davesh Maulik, Ananth N. Shankar, Yunqing Tang

TL;DR

The paper proves that for a generically ordinary non-isotrivial family of K3 surfaces over a curve in characteristic p ≥ 5, the geometric Picard rank jumps at infinitely many closed points when the generic Picard lattice discriminant is prime to p. It reframes the moduli problem in terms of GSpin Shimura varieties and analyzes special divisors Z(m) via Borcherds lifts, while controlling local contributions at supersingular points through F-crystal methods and lattice-decay arguments. These local-to-global bounds yield infinitely many intersections of a curve with special divisors, and the authors leverage this to establish the ordinary Hecke orbit conjecture for GSpin (and some unitary) Shimura varieties, proving Zariski-density of ordinary Hecke orbits. The results extend and complement previous work on Noether–Lefschetz/Lift-type phenomena, providing a robust framework for understanding Picard-jump phenomena and Hecke-orbit dynamics in positive characteristic.

Abstract

Let $\mathscr{X} \rightarrow C$ be a non-isotrivial and generically ordinary family of K3 surfaces over a proper curve $C$ in characteristic $p \geq 5$. 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 $\mathcal{S}$ 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 $\mathcal{S}_{\overline{\mathbb{F}}_p}$ intersects special divisors of $\mathcal{S}_{\overline{\mathbb{F}}_p}$ 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 $\mathcal{S}_{\overline{\mathbb{F}}_p}$ have Zariski-dense Hecke orbits. We also deduce the ordinary Hecke orbit conjecture for certain families of unitary Shimura varieties.

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

TL;DR

The paper proves that for a generically ordinary non-isotrivial family of K3 surfaces over a curve in characteristic p ≥ 5, the geometric Picard rank jumps at infinitely many closed points when the generic Picard lattice discriminant is prime to p. It reframes the moduli problem in terms of GSpin Shimura varieties and analyzes special divisors Z(m) via Borcherds lifts, while controlling local contributions at supersingular points through F-crystal methods and lattice-decay arguments. These local-to-global bounds yield infinitely many intersections of a curve with special divisors, and the authors leverage this to establish the ordinary Hecke orbit conjecture for GSpin (and some unitary) Shimura varieties, proving Zariski-density of ordinary Hecke orbits. The results extend and complement previous work on Noether–Lefschetz/Lift-type phenomena, providing a robust framework for understanding Picard-jump phenomena and Hecke-orbit dynamics in positive characteristic.

Abstract

Let be a non-isotrivial and generically ordinary family of K3 surfaces over a proper curve in characteristic . We prove that the geometric Picard rank jumps at infinitely many closed points of . More generally, suppose that we are given the canonical model of a Shimura variety of orthogonal type, associated to a lattice of signature that is self-dual at . We prove that any generically ordinary proper curve in intersects special divisors of 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 have Zariski-dense Hecke orbits. We also deduce the ordinary Hecke orbit conjecture for certain families of unitary Shimura varieties.

Paper Structure

This paper contains 15 sections, 35 theorems, 52 equations.

Table of Contents

  1. Introduction
  2. Families of K3 surfaces
  3. The Hecke orbit conjecture.
  4. Outline of the proof of Theorem \ref{['thm_int']}
  5. Contributions from supersingular points
  6. Outline of proof of Theorem \ref{['thm_Hecke']}
  7. Previous work
  8. Organization of paper
  9. GSpin Shimura varieties and special divisors
  10. Lattice decay statements and heuristics
  11. The $F$-crystal $\mathbf{L}_{\mathrm{cris}}$ on local deformation spaces of supersingular points
  12. Decay for non-superspecial supersingular points
  13. Decay for superspecial points
  14. Proof of \ref{['thm_int']}
  15. Application to the Hecke orbit problem

Key Result

Theorem 1.1

Let $C/\overline{\mathbb{F}}_p$ denote a smooth proper curve where $p \geq 5$ is a prime number, and let $\mathscr{X}\rightarrow C$ denote a generically ordinary non-isotrivial family of K3 surfaces. Suppose that the discriminantNote that the Picard lattice of a K3 surface is equipped with a non-deg

Theorems & Definitions (77)

  • Theorem 1.1
  • Theorem 1.2
  • Conjecture 1.3: Chai--Oort
  • Theorem 1.4
  • Definition 2.3: AGHMP
  • Remark 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 3.1
  • Theorem 3.2
  • ...and 67 more