Hyperelliptic curves mapping to abelian varieties and applications to Beilinson's conjecture for zerocycles
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 nonisomorphic 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 zerocycles $\text{CH}_0(A)$. We use these results to give some progress towards Beilinson's conjecture for zerocycles, which predicts that for a smooth projective variety $X$ over $\overline{Q}$ the kernel of the Albanese map of $X$ is zero.
 Publication:

arXiv eprints
 Pub Date:
 September 2023
 DOI:
 10.48550/arXiv.2309.06361
 arXiv:
 arXiv:2309.06361
 Bibcode:
 2023arXiv230906361G
 Keywords:

 Mathematics  Algebraic Geometry
 EPrint:
 27 pages. The statement of Theorem 1.3 (formerly Theorem 1.4) has been strengthened, and its proof in Section 3 has been changed significantly. Intersection theory computations have been moved to an appendix, and the introduction has been reorganized