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Motivic Modularity of CM K3 Surfaces

Rikuto Ito

TL;DR

The paper establishes motivic modularity for CM K3 surfaces: for a CM K3 surface $X$ over a number field $k$ with CM by a field $E$ and $k\supset E$, there exists a unique locally constant map $u$ giving algebraic Hecke characters $\chi_{\tau}$ whose associated $l$-adic characters diagonalize the transcendental Galois action. It proves the $l$-adic representation on the transcendental lattice is semisimple and decomposes as a direct sum of one-dimensional pieces corresponding to the $\chi_{\tau}$, leading to a factorization of the $L$-function $L(s, T_l(X_{\bar{k}})\otimes \mathbb{Q}_l)$ as a product of Hecke $L$-functions $\prod_{\tau}L(s,\chi_{\tau})$. The construction hinges on Rizov’s main CM theorem, linking the Galois action to a CM torus and class field theory, and yields explicit base-field and character data for CM K3 surfaces, extending prior partial results. This provides concrete a priori descriptions of the CM data associated to CM K3 surfaces and connects Galois representations with automorphic data in a precise, factorized form.

Abstract

Piatetski-Shapiro and Shafarevich proved the L-functions of K3 surfaces of CM type are expressed as the product of some Hecke L-functions by changing their base fields. In this paper, the aurhor gives the explicit description of these Hecke characters and base fields.

Motivic Modularity of CM K3 Surfaces

TL;DR

The paper establishes motivic modularity for CM K3 surfaces: for a CM K3 surface over a number field with CM by a field and , there exists a unique locally constant map giving algebraic Hecke characters whose associated -adic characters diagonalize the transcendental Galois action. It proves the -adic representation on the transcendental lattice is semisimple and decomposes as a direct sum of one-dimensional pieces corresponding to the , leading to a factorization of the -function as a product of Hecke -functions . The construction hinges on Rizov’s main CM theorem, linking the Galois action to a CM torus and class field theory, and yields explicit base-field and character data for CM K3 surfaces, extending prior partial results. This provides concrete a priori descriptions of the CM data associated to CM K3 surfaces and connects Galois representations with automorphic data in a precise, factorized form.

Abstract

Piatetski-Shapiro and Shafarevich proved the L-functions of K3 surfaces of CM type are expressed as the product of some Hecke L-functions by changing their base fields. In this paper, the aurhor gives the explicit description of these Hecke characters and base fields.

Paper Structure

This paper contains 9 sections, 15 theorems, 44 equations.

Table of Contents

  1. Introduction
  2. Main theorem
  3. Preliminaries
  4. Abelian $l$-adic representations
  5. Mumford-Tate groups
  6. K3 surfaces with CM
  7. The proof of Theorem 1.1
  8. The main theorem of complex multiplications by Rizov
  9. Construction of Hecke characters

Key Result

Theorem 1.1

Let $X$ be a K3 surface defined over a number field $k\subset \mathbb{C}$. Suppose that $X$ has CM over $k$ by the CM field $E$, i.e., $X_{\mathbb{C}}$ is of CM type and $k\supset E$. Then there exists the unique locally constant homomorphism $u : \mathbb{A}^{\times}_{k} \to E^{\times}\subset \mathb is an algebraic Hecke character, where ${\rm Nm}_{k/E} : \mathbb{A}^{\times}_{k}\to \mathbb{A}^{\ti

Theorems & Definitions (46)

  • Theorem 1.1
  • Remark 1.2
  • Definition 2.1
  • Example 2.2
  • Theorem 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Definition 2.6
  • Remark 2.7
  • ...and 36 more