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On the component group of the algebraic monodromy group of a $K3$ surface

Andreas-Stephan Elsenhans, Jörg Jahnel

Abstract

We provide a lower bound for the number of components of the algebraic monodromy group in the situation of a $K3$ surface over a number field $k$. In the CM case, our bound is sharp. As an application, we describe, in the case of CM, the jump character \cite[Definition~2.4.6]{CEJ} entirely in terms of the endomorphism field and the geometric Picard rank.

On the component group of the algebraic monodromy group of a $K3$ surface

Abstract

We provide a lower bound for the number of components of the algebraic monodromy group in the situation of a surface over a number field . In the CM case, our bound is sharp. As an application, we describe, in the case of CM, the jump character \cite[Definition~2.4.6]{CEJ} entirely in terms of the endomorphism field and the geometric Picard rank.
Paper Structure (8 sections, 18 theorems, 71 equations, 4 figures, 1 table)

This paper contains 8 sections, 18 theorems, 71 equations, 4 figures, 1 table.

Table of Contents

  1. Introduction
  2. Technical prerequisites
  3. The various actions of the endomorphism field
  4. The actions of the endomorphism field and the Galois group on algebraic de Rham and $l$-adic cohomologies---The main results
  5. The fundamental lemma on the CM case
  6. A digression: the field of definition of the conjectural correspondence describing real or complex multiplication
  7. Results relying on the Hodge conjecture
  8. Examples

Key Result

Theorem 2.7

Let $k$ be a number field and let $l$ and $l'$ be two prime numbers. Suppose that the two rational representations ${\varrho\colon \mathop{\text{\rm Gal}}\nolimits(\overline{k}/k) \to \mathop{\text{\rm GL}}\nolimits_n(\overline{\mathbbm Q}_l)}$ and ${\varrho'\colon \mathop{\text{\rm Gal}}\nolimits(\ to the component groups have the same kernel. Proof.This is shown in Se81. [-22pt]0pt1pt□ $\squar

Figures (4)

  • Figure 1: Theoretical and experimental trace distributions for the $K3$ surface $X_2$ over $k={\mathbbm Q}$ of geometric Picard rank $18$ having CM by ${E={\mathbbm Q}(\zeta_5)}$
  • Figure 2: Theoretical and experimental trace distributions for the $K3$ surface $X_3$ over $k={\mathbbm Q}$ of geometric Picard rank $18$ having CM by ${E={\mathbbm Q}(\sqrt{2},i)}$
  • Figure 3: Theoretical and experimental trace distributions for the $K3$ surface $X_5$ over a non-normal cubic number field $k$ of geometric Picard rank $16$ having CM by ${E=k(i)}$
  • Figure 4: Theoretical and experimental trace distributions for the $K3$ surface $X_6$ over $k={\mathbbm Q}$ of geometric Picard rank $16$ having RM by ${E={\mathbbm Q}(\sqrt{3})}$

Theorems & Definitions (38)

  • Remark 1.2
  • Remark 1.3
  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Example 2.4
  • Definition 2.5
  • Theorem 2.7: J.-P. Serre
  • Corollary 2.8
  • Lemma 2.12: Descent for linear maps
  • ...and 28 more