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arxiv: 2402.12588 · v2 · pith:J7GAJTJLnew · submitted 2024-02-19 · 🧮 math.AG

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

Evangelia Gazaki , Jonathan Love This is my paper

Pith reviewed 2026-05-24 03:54 UTC · model grok-4.3

classification 🧮 math.AG
keywords K3 surfaceszero-cyclesChow groupsKummer surfacesp-adic fieldsBrauer-Manin obstructionlocal-to-global principles
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The pith

For geometrically Kummer K3 surfaces over p-adic fields from products of elliptic curves, the group of degree-zero zero-cycles decomposes as a divisible group plus a finite group.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes a structure theorem for the Chow group of zero-cycles of degree zero on a specific class of K3 surfaces over p-adic fields. These surfaces become Kummer surfaces associated to abelian surfaces isogenous to products of elliptic curves after base change to an algebraic closure. Under assumptions on the reduction types of those elliptic curves, the group splits as a divisible part plus a finite part. This confirms a conjecture of Raskind-Spiess and Colliot-Thélène in full for the first time among K3 surfaces. The results also address a local-to-global principle for the Brauer-Manin obstruction on zero-cycles, including examples over number fields and some unconditional cases.

Core claim

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, the Chow group A0(X) of zero-cycles of degree 0 on X is the direct sum of a divisible group and a finite group. This class includes diagonal quartic surfaces, and in good ordinary reduction many cases allow the finite summand to be determined completely.

What carries the argument

The geometric isomorphism of X over the algebraic closure to the Kummer surface of an abelian surface isogenous to a product of two elliptic curves, together with the reduction type assumptions that control the structure of A0(X).

If this is right

  • This proves the Raskind-Spiess and Colliot-Thélène conjecture for this class of K3 surfaces.
  • In cases of good ordinary reduction the finite summand of A0(X) can be completely determined in many cases.
  • Examples exist of Kummer surfaces over number fields where places of good ordinary reduction contribute nontrivially to the Brauer set for zero-cycles of degree 0.
  • Unconditional local-to-global principles for zero-cycles of degree 0 can be proved in some cases for these K3 surfaces.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The decomposition might extend to K3 surfaces that are not geometrically Kummer but share similar reduction or isogeny properties.
  • Explicit computation of the finite summand on a concrete diagonal quartic could reveal whether its order follows a pattern tied to the elliptic curve conductors.
  • The local-to-global evidence could be tested numerically on other K3 surfaces over number fields by checking whether the Brauer set matches the adelic points for zero-cycles.
  • Similar control via reduction types might produce structure results for Chow groups of zero-cycles on abelian surfaces or their quotients in other characteristics.

Load-bearing premise

X must be isomorphic over the algebraic closure to the Kummer surface of an abelian surface isogenous to a product of two elliptic curves whose reduction types satisfy the stated conditions.

What would settle it

An explicit diagonal quartic surface over a p-adic field meeting the reduction assumptions where the Chow group A0(X) contains an element of infinite order not divisible by arbitrarily high powers of any prime.

read the original 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-Th\'{e}l\`{e}ne 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-Th\'{e}lene, 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 2 minor

Summary. The paper proves that if X is a K3 surface over a p-adic field k that is geometrically isomorphic to the Kummer surface of an abelian surface A isogenous to a product E1 × E2 of elliptic curves, then under explicit assumptions on the reduction types of E1 and E2, the Chow group A0(X) of degree-zero zero-cycles decomposes as the direct sum of a divisible group and a finite group. This establishes the Raskind–Spiess / Colliot-Thélène conjecture for this class of K3 surfaces (including diagonal quartics). The paper further studies the associated local-to-global conjecture for zero-cycles, exhibiting cases in which the Brauer–Manin obstruction is nontrivial at places of good ordinary reduction and cases in which an unconditional local-to-global principle holds.

Significance. If the stated results hold, the work supplies the first complete verification of the Raskind–Spiess / Colliot-Thélène conjecture for any class of K3 surfaces and supplies the first unconditional evidence for the Colliot-Thélène–Sansuc–Kato–Saito local-to-global conjecture in the K3 setting. The explicit control of the finite summand under good ordinary reduction and the concrete examples over number fields are additional strengths.

minor comments (2)
  1. The abstract states that the result is “the first instance for K3 surfaces when this conjecture is proved in full.” A brief sentence in the introduction comparing the present hypotheses with earlier partial results on K3 surfaces would help readers situate the advance.
  2. In the local-to-global section, the precise definition of the Brauer set for zero-cycles of degree zero is invoked without an explicit reference to the relevant earlier paper; adding the citation at first use would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive report, which accurately summarizes the main results on the decomposition of A0(X) for geometrically Kummer K3 surfaces and the local-to-global results for zero-cycles. We are pleased that the referee recommends acceptance.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper establishes a structure theorem for A0(X) on geometrically Kummer K3 surfaces via reduction to the product case using isogenies, control of Galois cohomology and cycle maps under explicit reduction hypotheses on elliptic factors, and application of known results on Chow groups of elliptic curves and Kummer quotients. No self-definitional equivalences, fitted inputs renamed as predictions, load-bearing self-citations reducing the central claim, or ansatzes smuggled via citation appear in the derivation chain. The proof is self-contained against external benchmarks in algebraic geometry and Chow theory.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard facts about Chow groups of K3 surfaces, the Kummer construction, and reduction-type hypotheses on elliptic curves; no free parameters or new invented entities are introduced in the abstract.

axioms (2)
  • standard math Standard properties of the Chow group A0 of a K3 surface and of Kummer surfaces associated to abelian surfaces
    Invoked to obtain the direct-sum decomposition under the stated geometric hypotheses.
  • domain assumption Assumptions on reduction types of the elliptic curve factors of A
    Explicitly required in the abstract to prove the structure of A0(X).

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Reference graph

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