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

The Kawazumi-Zhang invariant phi 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 phi in the non-separating degeneration limit, it is shown that phi 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 phi near the non-separating node, gives full control on the asymptotics of phi in the various degeneration limits, and provides an efficient numerical procedure to evaluate phi to arbitrary accuracy. It also reveals a mock-type holomorphic Siegel modular form of weight -2 underlying phi. 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. (c) 2016 Elsevier Inc. All rights reserved.