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Grothendieck's period conjecture for Kummer surfaces of self-product CM type

Daiki Kawabe

Abstract

We show that the Grothendieck period conjecture holds for the Kummer surface associated with the square of a CM elliptic curve. This means that the period isomorphism is dense in the torsor of motivic periods. In other words, the isomorphism is dense in the torsor of motivated periods, and motivated classes on powers of the surface are algebraic. The point is that the motive has a non-trivial transcendental part, but belongs to the Tannakian category generated by the motive of a CM elliptic curve.

Grothendieck's period conjecture for Kummer surfaces of self-product CM type

Abstract

We show that the Grothendieck period conjecture holds for the Kummer surface associated with the square of a CM elliptic curve. This means that the period isomorphism is dense in the torsor of motivic periods. In other words, the isomorphism is dense in the torsor of motivated periods, and motivated classes on powers of the surface are algebraic. The point is that the motive has a non-trivial transcendental part, but belongs to the Tannakian category generated by the motive of a CM elliptic curve.
Paper Structure (7 sections, 16 theorems, 10 equations)

This paper contains 7 sections, 16 theorems, 10 equations.

Table of Contents

  1. Introduction
  2. Motivated version of the Grothendieck period conjecture
  3. Motivated Galois groups
  4. Mumford-Tate groups
  5. The power of an elliptic curve
  6. Surfaces with $b_{1} = b_{3} = 0$
  7. Proof of the main theorem and remarks

Key Result

Theorem 1.3

Let $A$ be an abelian surface over $\overline{\mathbb{Q}}$ isogenous to the square $E^{2}$ of a CM elliptic curve. Let $\mathrm{Km}(A)$ be the Kummer surface over $\overline{\mathbb{Q}}$ associated with $A$. Then the GPC holds for $\mathrm{Km}(A)$.

Theorems & Definitions (36)

  • Conjecture 1.1
  • Remark 1.2
  • Theorem 1.3
  • Conjecture 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • proof
  • Lemma 2.4
  • proof
  • ...and 26 more