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Local and local-to-global Principles for zero-cycles on geometrically Kummer $K3$ surfaces

Evangelia Gazaki, Jonathan Love

Abstract

Let $X$ be a $K3$ surface over a $p$-adic field $k$ such that for some abelian surface $A$ isogenous to a product of two elliptic curves, there is an isomorphism over the algebraic closure of $k$ between $X$ and the Kummer surface associated to $A$. Under some assumptions on the reduction types of the elliptic curve factors of $A$, we prove that the Chow group $A_0(X)$ of zero-cycles of degree $0$ on $X$ is the direct sum of a divisible group and a finite group. This proves a conjecture of Raskind and Spiess and of Colliot-Thélène and it is the first instance for $K3$ surfaces when this conjecture is proved in full. This class of $K3$'s includes, among others, the diagonal quartic surfaces. In the case of good ordinary reduction we describe many cases when the finite summand of $A_0(X)$ can be completely determined. Using these results, we explore a local-to-global conjecture of Colliot-Thélene, Sansuc, Kato and Saito which, roughly speaking, predicts that the Brauer-Manin obstruction is the only obstruction to Weak Approximation for zero-cycles. We give examples of Kummer surfaces over a number field $F$ where the ramified places of good ordinary reduction contribute nontrivially to the Brauer set for zero-cycles of degree $0$ and we describe cases when an unconditional local-to-global principle can be proved, giving the first unconditional evidence for this conjecture in the case of $K3$ surfaces.

Local and local-to-global Principles for zero-cycles on geometrically Kummer $K3$ surfaces

Abstract

Let be a surface over a -adic field such that for some abelian surface isogenous to a product of two elliptic curves, there is an isomorphism over the algebraic closure of between and the Kummer surface associated to . Under some assumptions on the reduction types of the elliptic curve factors of , we prove that the Chow group of zero-cycles of degree on is the direct sum of a divisible group and a finite group. This proves a conjecture of Raskind and Spiess and of Colliot-Thélène and it is the first instance for surfaces when this conjecture is proved in full. This class of 's includes, among others, the diagonal quartic surfaces. In the case of good ordinary reduction we describe many cases when the finite summand of can be completely determined. Using these results, we explore a local-to-global conjecture of Colliot-Thélene, Sansuc, Kato and Saito which, roughly speaking, predicts that the Brauer-Manin obstruction is the only obstruction to Weak Approximation for zero-cycles. We give examples of Kummer surfaces over a number field where the ramified places of good ordinary reduction contribute nontrivially to the Brauer set for zero-cycles of degree and we describe cases when an unconditional local-to-global principle can be proved, giving the first unconditional evidence for this conjecture in the case of surfaces.
Paper Structure (15 sections, 23 theorems, 68 equations)

This paper contains 15 sections, 23 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Results over $p$-adic fields
  3. Weak Approximation for zero-cycles
  4. Notation
  5. Acknowledgment
  6. The local Conjecture
  7. Background and notation
  8. Kummer surfaces
  9. Proof of \ref{['localtheointro']}
  10. Diagonal Quartic Surfaces
  11. Explicit local structure
  12. Unramified Divisibility
  13. Computing the non-divisible subgroup and relation to the Brauer group
  14. Weak Approximation for zero-cycles
  15. The role of good reduction places

Key Result

Theorem 1

(cf. padic1) Let $k$ be a finite extension of the $p$-adic field $\mathbb{Q}\xspace_p$. Let $X$ be a $K3$ surface over $k$. Suppose there exists a finite extension $K/k$ such that the base change $X_K:=X\otimes_k K$ becomes isomorphic to the Kummer surface $\mathop{\mathrm{Kum}}\nolimits(A)$ associa

Theorems & Definitions (57)

  • Conjecture 1
  • Theorem 1
  • Corollary 1
  • Conjecture 2
  • Definition 1
  • Theorem 2
  • Proposition 1
  • Example 1
  • Example 2
  • Example 3
  • ...and 47 more