A Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces

TL;DR

This paper establishes a Theta lift representation for the Kawazumi-Zhang and Faltings invariants of genus-two Riemann surfaces, enabling precise analysis of their properties and asymptotics.

Contribution

It identifies the Kawazumi-Zhang invariant as a Theta lift of a weak Jacobi form, providing explicit Fourier-Jacobi expansion, asymptotic control, and a numerical evaluation method.

Findings

Kawazumi-Zhang invariant is a Theta lift of a weak Jacobi form.

Provides explicit Fourier-Jacobi expansion near the non-separating node.

Reveals a mock-type holomorphic Siegel modular form underlying the invariant.

Abstract

The Kawazumi-Zhang invariant $\varphi$ for compact genus-two Riemann surfaces was recently shown to be a eigenmode of the Laplacian on the Siegel upper half-plane, away from the separating degeneration divisor. Using this fact and the known behavior of $\varphi$ in the non-separating degeneration limit, it is shown that $\varphi$ is equal to the Theta lift of the unique (up to normalization) weak Jacobi form of weight $-2$. This identification provides the complete Fourier-Jacobi expansion of $\varphi$ near the non-separating node, gives full control on the asymptotics of $\varphi$ in the various degeneration limits, and provides a efficient numerical procedure to evaluate $\varphi$ to arbitrary accuracy. It also reveals a mock-type holomorphic Siegel modular form of weight $-2$ underlying $\varphi$. From the general relation between the Faltings invariant, the Kawazumi-Zhang invariant and the discriminant for hyperelliptic Riemann surfaces, a Theta lift representation for the Faltings invariant in genus two readily follows.