Algebraic Cycles and values of Green's functions

TL;DR

This paper constructs special algebraic cycles in motivic cohomology of Abelian surfaces over number or function fields, using them to establish algebraicity of higher Green's function values and relating to Gross-Kohnen-Zagier conjecture.

Contribution

It introduces new indecomposable motivic cycles on Abelian surfaces and applies them to prove algebraicity results for higher Green's functions, connecting to recent conjectures.

Findings

Construction of indecomposable motivic cycles on Abelian surfaces.

Proof of algebraicity of higher Green's function values.

Formulation of a conjecture linking to Gross-Kohnen-Zagier work.

Abstract

We construct indecomposable cycles in the motivic cohomology group $H^3_{{\mathcal M}}(A,{\mathbb Q}(2))$ where $A$ is an Abelian surface over a number field or the function field of a base. When $A$ is the self product of the universal elliptic curve over a modular curve, these cycles can be used to prove algebraicity results for values of higher Green's functions, similar to a conjecture of Gross, Kohnen and Zagier. We formulate a conjecture which relates our work with the recent work of Bruinier-Ehlen-Yang on the conjecture of Gross-Kohnen-Zagier.