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Potential Automorphy of K3 Surfaces with Large Picard Rank

Chao Gu

TL;DR

The work develops a robust framework to prove potential modularity for K3 surfaces with large Picard rank by leveraging $\mathrm{GSp}_4$-type abelian varieties and their compatible systems. It combines a lattice of tools—Moret-Bailly globalization, Serre–Tate lifts, Shimura/PEL moduli spaces, and modularity lifting theorems—to transport automorphy from abelian-type motives to the transcendental motives of K3 surfaces. The main achievement is establishing potential modularity for K3 surfaces with geometric Picard rank at least 17, with extended results for ranks 18–20, and a detailed pathway through real and fake $\mathrm{GSp}_4$-type cases, descent data, and induced representations. This yields the Hasse–Weil conjecture for many such K3 surfaces and illuminates how high Picard rank constrains the monodromy to enable modularity methods in higher dimensions.

Abstract

The first part of this paper studied $\mathrm{GSp}_4$-type abelian varieties and the corresponding compatible systems of $\mathrm{GSp}_4$ representations. Techniques in \cite{BCGP} are applied to show that one can prove the potential modularity of these abelian varieties and compatible systems under some conditions that guarantee a sufficient amount of good primes. Then, in the second part, we use the potential modularity theorems to prove that K3 surfaces over totally real field $F$ with Picard rank $\ge 17$ are potentially modular.

Potential Automorphy of K3 Surfaces with Large Picard Rank

TL;DR

The work develops a robust framework to prove potential modularity for K3 surfaces with large Picard rank by leveraging -type abelian varieties and their compatible systems. It combines a lattice of tools—Moret-Bailly globalization, Serre–Tate lifts, Shimura/PEL moduli spaces, and modularity lifting theorems—to transport automorphy from abelian-type motives to the transcendental motives of K3 surfaces. The main achievement is establishing potential modularity for K3 surfaces with geometric Picard rank at least 17, with extended results for ranks 18–20, and a detailed pathway through real and fake -type cases, descent data, and induced representations. This yields the Hasse–Weil conjecture for many such K3 surfaces and illuminates how high Picard rank constrains the monodromy to enable modularity methods in higher dimensions.

Abstract

The first part of this paper studied -type abelian varieties and the corresponding compatible systems of representations. Techniques in \cite{BCGP} are applied to show that one can prove the potential modularity of these abelian varieties and compatible systems under some conditions that guarantee a sufficient amount of good primes. Then, in the second part, we use the potential modularity theorems to prove that K3 surfaces over totally real field with Picard rank are potentially modular.

Paper Structure

This paper contains 20 sections, 40 theorems, 85 equations.

Table of Contents

  1. Backgrounds
  2. Introduction and Notations
  3. Compatible System of $\mathrm{GSp}_4$ representations
  4. Abelian Varieties of $\mathrm{GSp}_4$-type
  5. Real $\mathrm{GSp}_4$ Type Abelian Varieties
  6. Fake $\mathrm{GSp}_4$ Type Abelian Varieties
  7. Abelian Varieties with Descent Data
  8. Large Image Property
  9. The Large Image Hypothesis
  10. Distinguished Ordinary Primes
  11. The Serre-Tate Canonical Lift
  12. Moduli Spaces of $\mathrm{GSp}_4$ type Abelian Varieties
  13. Construction of the Shimura Variety
  14. Proof of Theorem \ref{['MB']}
  15. Potential Modularity of $\mathrm{GSp}_4$ type Abelian Varieties
  16. ...and 5 more sections

Key Result

Theorem 1.1

Suppose that the geometric Picard rank of $X$ is $\rho\ge 17$. Then $T$ is potentially modular, and the Hasse-Weil conjecture holds for $X$.

Theorems & Definitions (79)

  • Theorem 1.1
  • Example 1.2
  • Definition 1.3
  • Proposition 1.4
  • Definition 1.5
  • Lemma 1.6
  • proof
  • Definition 1.7
  • Definition 1.8
  • Definition 2.1
  • ...and 69 more