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

TL;DR

This paper constructs numerous hyperelliptic curves on certain abelian surfaces and uses them to make progress on Beilinson's conjecture concerning zero-cycles and the Albanese kernel.

Contribution

It describes a large family of hyperelliptic curves mapping into abelian surfaces isogenous to products of elliptic curves, advancing understanding of zero-cycles and Beilinson's conjecture.

Findings

Constructed infinitely many hyperelliptic curves of various genera on abelian surfaces

Established rational equivalences in the Chow group of zero-cycles

Provided evidence towards Beilinson's conjecture for zero-cycles

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.