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A Filtration of the Chow Group of Zero-Cycles for a Product of Curves and an Abelian Variety

Thomas Jaklitsch

TL;DR

The paper constructs a descending integral filtration F^r CH_0(X) for X ≅ C_1×...×C_d×A (A abelian, C_i curves with k-points), linking its successive quotients to quotients of Somekawa K-groups S_r(k; underline{J_1}×...×underline{J_d}×A). It defines Φ_r to map CH_0(X) into these K-groups and builds an inverse-like Ψ_r, establishing Φ_r∘Ψ_r = r! so that the quotients are described after inverting r!. The results extend known filtrations for products of curves and for abelian varieties, provide explicit generators and relations via K-groups, and include a finiteness statement F^r CH_0(X)⊗Q = 0 for r exceeding the dimension, with a detailed genus-2 example illustrating the method and its arithmetic significance.

Abstract

In this paper we define a descending filtration on the Chow group of zero cycles for varieties of the form $A \times C_1 \times \cdots \times C_d$ where $A$ is an abelian variety and each $C_i$ is a smooth projective curve. We give explicit generators and relations for the successive quotients of this filtration by showing that they can be described by Somekawa K-groups. This extends the work of Raskind and Spiess who proved this result for products of curves and Gazaki who proved this for abelian varieties.

A Filtration of the Chow Group of Zero-Cycles for a Product of Curves and an Abelian Variety

TL;DR

The paper constructs a descending integral filtration F^r CH_0(X) for X ≅ C_1×...×C_d×A (A abelian, C_i curves with k-points), linking its successive quotients to quotients of Somekawa K-groups S_r(k; underline{J_1}×...×underline{J_d}×A). It defines Φ_r to map CH_0(X) into these K-groups and builds an inverse-like Ψ_r, establishing Φ_r∘Ψ_r = r! so that the quotients are described after inverting r!. The results extend known filtrations for products of curves and for abelian varieties, provide explicit generators and relations via K-groups, and include a finiteness statement F^r CH_0(X)⊗Q = 0 for r exceeding the dimension, with a detailed genus-2 example illustrating the method and its arithmetic significance.

Abstract

In this paper we define a descending filtration on the Chow group of zero cycles for varieties of the form where is an abelian variety and each is a smooth projective curve. We give explicit generators and relations for the successive quotients of this filtration by showing that they can be described by Somekawa K-groups. This extends the work of Raskind and Spiess who proved this result for products of curves and Gazaki who proved this for abelian varieties.

Paper Structure

This paper contains 14 sections, 19 theorems, 184 equations.

Table of Contents

  1. Introduction
  2. Outline
  3. Notation
  4. Acknowledgments
  5. Definitions
  6. Somekawa K-groups
  7. Chow Groups
  8. The Filtration
  9. Proof of the Main Theorem
  10. Definition of $\Psi_r$ and Proof of Theorem \ref{['thm: main']}
  11. Proof of Key Lemmas
  12. Finiteness of the Filtration
  13. Specialization Maps
  14. Example: Chow Group of a Genus Two Curve and its Jacobian

Key Result

Theorem 1.1

Let $A$ be an abelian variety and let $C_1, \ldots, C_d$ be smooth projective geometrically irreducible curves such that $C(k) \ne \emptyset$. Then there exists an isomorphism

Theorems & Definitions (50)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 1
  • Remark 1
  • Definition 2
  • Proposition 1
  • proof
  • Definition 3
  • Proposition 2
  • proof
  • ...and 40 more