期刊
JOURNAL OF NUMBER THEORY
卷 163, 期 -, 页码 520-541出版社
ACADEMIC PRESS INC ELSEVIER SCIENCE
DOI: 10.1016/j.jnt.2015.12.021
关键词
Theta lift; Kawazumi-Zhang invariant; Faltings invariant; Holomorphic prepotential; Mock Siegel modular form
类别
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.
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