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

The Kawazumi–Zhang invariant φ for compact genus-two Riemann surfaces was recently shown to be an eigenmode of the Laplacian on the Siegel upper half-plane, away from the separating degeneration divisor. Using this fact and the known behavior of φ in the non-separating degeneration limit, it is shown that φ 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 φ near the non-separating node, gives full control on the asymptotics of φ in the various degeneration limits, and provides an efficient numerical procedure to evaluate φ to arbitrary accuracy. It also reveals a mock-type holomorphic Siegel modular form of weight −2 underlying φ. 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.