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Hyperelliptic curves mapping to abelian varieties and applications to Beilinson's conjecture for zero-cycles

Evangelia Gazaki, Jonathan R. Love

Abstract

Let $A$ be an abelian surface over an algebraically closed field $\overline{k}$ with an embedding $\overline{k}\hookrightarrow\mathbb{C}$. When $A$ is isogenous to a product of elliptic curves, we describe a large collection of pairwise non-isomorphic hyperelliptic curves mapping birationally into $A$. For infinitely many integers $g\geq 2$, this collection has infinitely many curves of genus $g$, and no two curves in the collection have the same image under any isogeny from $A$. Using these hyperelliptic curves, we find many rational equivalences in the Chow group of zero-cycles $\text{CH}_0(A)$. We use these results to give some progress towards Beilinson's conjecture for zero-cycles, which predicts that for a smooth projective variety $X$ over $\overline{\mathbb{Q}}$ the kernel of the Albanese map of $X$ is zero.

Hyperelliptic curves mapping to abelian varieties and applications to Beilinson's conjecture for zero-cycles

Abstract

Let be an abelian surface over an algebraically closed field with an embedding . When is isogenous to a product of elliptic curves, we describe a large collection of pairwise non-isomorphic hyperelliptic curves mapping birationally into . For infinitely many integers , this collection has infinitely many curves of genus , and no two curves in the collection have the same image under any isogeny from . Using these hyperelliptic curves, we find many rational equivalences in the Chow group of zero-cycles . We use these results to give some progress towards Beilinson's conjecture for zero-cycles, which predicts that for a smooth projective variety over the kernel of the Albanese map of is zero.
Paper Structure (22 sections, 27 theorems, 96 equations, 1 table)

This paper contains 22 sections, 27 theorems, 96 equations, 1 table.

Table of Contents

  1. Introduction
  2. Progress towards Beilinson's Conjecture
  3. Acknowledgement
  4. Rational Equivalences on Abelian Surfaces
  5. Hyperelliptic points
  6. Zero-cycles on Abelian Surfaces
  7. Some Progress towards Beilinson's Conjecture
  8. Producing curves in abelian surfaces
  9. Kummer surface as elliptic fibration
  10. Bounding the genus of $H$
  11. Improving the lower bound
  12. Compatible maps and isogenies
  13. Proof of \ref{['infinite_curves']}
  14. Explicit Examples
  15. Degenerate sections
  16. ...and 7 more sections

Key Result

Theorem 1

(cf. abelsurface1) Let $A$ be an abelian surface over an algebraically closed field $\overline{k}\xspace$ and let $a,b\in A(\overline{k}\xspace)$. Suppose there exist nonzero integers $m,m'$ such that $a$, $b$, and $ma+m'b$ are hyperelliptic points. Let $B_{a,b}$ be the divisible hull of the subgrou vanishes in the Albanese kernel $F^2(A)$.

Theorems & Definitions (62)

  • Conjecture 1
  • Theorem 1
  • Theorem 2
  • Definition 1
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Theorem 3
  • Corollary 1
  • ...and 52 more