Higher Chow Cycles on an Abelian Surface and a non-Archimedean analogue of the Hodge-D-conjecture

TL;DR

This paper constructs new indecomposable elements in the higher Chow group of a principally polarized Abelian surface over a non-Archimedean field, providing evidence for a non-Archimedean analogue of the Hodge-D-conjecture.

Contribution

It introduces a novel method to construct indecomposable elements in higher Chow groups of Abelian surfaces over non-Archimedean fields, advancing understanding of the non-Archimedean Hodge-D-conjecture.

Findings

Construction of new indecomposable elements in CH2(A,1)

Proof of the non-Archimedean Hodge-D-conjecture for certain Abelian surfaces

Extension of classical Humbert construction using recent deformation techniques

Abstract

We construct new indecomposable elements in the higher Chow group CH2(A,1) of a principally polarized Abelian surface over a non Archimedean local field, which generalize an element constructed by Collino. These elements are constructed using a generalization, due to Birkenhake and Wilhelm, of a classical construction of Humbert, along with some recent work of Bogomolov, Hassett and Tschinkel on deformations of rational curves on a K3 surface. They can be used to prove the non-Archimedean Hodge-D-conjecture - namely, the surjectivity of the boundary map in the localization sequence - in the case when the Abelian surface has good and ordinary reduction. This is a revised and updated version of an earlier preprint with the name `Abelian surfaces, Kummer surfaces and the non-Archimedean Hodge-D-conjecture.'