Siegel modular forms arising from higher Chow cycles

TL;DR

This paper establishes a connection between higher Chow cycles on generic abelian varieties and meromorphic Siegel modular forms, revealing a functorial relationship with degeneration processes.

Contribution

It introduces a novel construction linking higher Chow cycles to Siegel modular forms and demonstrates functoriality with respect to degeneration.

Findings

Higher Chow cycles induce meromorphic Siegel modular forms.

The construction respects degeneration via the Siegel operator.

The invariant has bounded singularity.

Abstract

We prove that the infinitesimal invariant of a higher Chow cycle of type (2,3-g) on a generic abelian variety of dimension g<4 gives rise to a meromorphic Siegel modular form of (virtual) weight Sym^{4}det^{-1} with bounded singularity, and that this construction is functorial with respect to rank 1 degeneration, namely the K-theory elevator for the cycle corresponds to the Siegel operator for the modular form.