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Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields

Ananth N. Shankar, Arul Shankar, Yunqing Tang, Salim Tayou

TL;DR

This work proves that a K3 surface $X$ over a number field $K$ with potentially good reduction everywhere exhibits infinitely many primes at which the geometric Picard rank jumps: $ ho(ar{X}_{ rak P})> ho(X_{ar{K}})$. Embedding the problem into the GSpin Shimura variety framework via the Kuga–Satake correspondence, the authors develop an arithmetic intersection theory for special divisors $ ext{Z}(m)$, pairing Green functions with heights and using the modularity of their generating series. They bound archimedean contributions through Bruinier’s Green functions and lattice-point counts (circle method) as well as non-archimedean contributions via deformations of special endomorphisms, culminating in a contradiction argument unless infinitely many primes admit a jump, which yields corollaries about rational curves and unirational specializations, and broader exceptional isogenies for Kuga–Satake abelian varieties and unitary Shimura-variety parametrizations. The approach integrates quadratic-form analytics, modularity, and height bounds to produce a robust, general mechanism for detecting exceptional reductions across families of K3-type motives. The results illuminate how Picard-rank fluctuations reflect deep arithmetic-geometric structures encoded in GSpin-Shimura settings, with broad consequences for abelian varieties and rational curves on K3 surfaces.

Abstract

Given a K3 surface $X$ over a number field $K$ with potentially good reduction everywhere, we prove that the set of primes of $K$ where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $X_{\overline{K}}$ has infinitely many rational curves or $X$ has infinitely many unirational specializations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field $K$ with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of $K$.

Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields

TL;DR

This work proves that a K3 surface over a number field with potentially good reduction everywhere exhibits infinitely many primes at which the geometric Picard rank jumps: . Embedding the problem into the GSpin Shimura variety framework via the Kuga–Satake correspondence, the authors develop an arithmetic intersection theory for special divisors , pairing Green functions with heights and using the modularity of their generating series. They bound archimedean contributions through Bruinier’s Green functions and lattice-point counts (circle method) as well as non-archimedean contributions via deformations of special endomorphisms, culminating in a contradiction argument unless infinitely many primes admit a jump, which yields corollaries about rational curves and unirational specializations, and broader exceptional isogenies for Kuga–Satake abelian varieties and unitary Shimura-variety parametrizations. The approach integrates quadratic-form analytics, modularity, and height bounds to produce a robust, general mechanism for detecting exceptional reductions across families of K3-type motives. The results illuminate how Picard-rank fluctuations reflect deep arithmetic-geometric structures encoded in GSpin-Shimura settings, with broad consequences for abelian varieties and rational curves on K3 surfaces.

Abstract

Given a K3 surface over a number field with potentially good reduction everywhere, we prove that the set of primes of where the geometric Picard rank jumps is infinite. As a corollary, we prove that either has infinitely many rational curves or has infinitely many unirational specializations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of .

Paper Structure

This paper contains 45 sections, 45 theorems, 167 equations.

Table of Contents

  1. Introduction
  2. Picard rank jumps for K3 surfaces
  3. Rational curves on K3 surfaces
  4. Exceptional splitting of abelian varieties
  5. GSpin Shimura varieties.
  6. Strategy of the proof
  7. Organization of the paper
  8. Acknowledgements
  9. Notations
  10. The GSpin Shimura varieties and their special divisors
  11. The GSpin Shimura variety
  12. The Kuga--Satake construction and K3 type motives in characteristic $0$
  13. Special divisors on $M$ over $\mathbb{Q}$
  14. Integral models
  15. Special endomorphisms and integral models of special divisors
  16. ...and 30 more sections

Key Result

Theorem 1.1

Let $X$ be a K3 surface over a number field $K$ and assume that $X$ admits, up to a finite extension of $K$, a projective smooth model $\mathcal{X}\rightarrow \mathrm{Spec}(\mathcal{O}_K)$. Then there are infinitely many finite places $\mathfrak{P}$ of $K$ such that $\rho(\mathcal{X}_{\overline{\mat

Theorems & Definitions (84)

  • Theorem 1.1
  • Conjecture 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.6
  • Corollary 1.7
  • Theorem 1.8
  • Proposition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 74 more