For an abelian surface A over an algebraically closed field k we describe a rich collection of rational equivalences in the kernel of the Albanese map of A arising from hyperelliptic curves. When the field k has characteristic zero and the abelian surface A is isogenous to a product of elliptic curves, we produce a very large collection of hyperelliptic curves mapping to A. We show that infinitely many of these curves have the same genus and the genera that appear tend to infinity. We use these results to obtain a substantial reduction to Beilinson's conjecture for zero-cycles when k is the algebraic closure of the rational numbers.
Hyperelliptic Curves mapping to Abelian Surfaces and applications to Beilinson's Conjecture for zero-cycles
Evangelia Gazaki, University of VirginiaAuthors: Evangelia Gazaki, Jonathan Love
2023 AWM Research Symposium
Recent Advances in Curves and Abelian Varieties [Organized by Renee Bell, Padmavathi Srinivasan, and Isabel Vogt]