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Holomorphic factorization of determinants of Laplacians on Riemann surfaces and a higher genus generalization of Kronecker’s first limit formula

By Andrew Mcintyre and Leon A. Takhtajan


For a family of compact Riemann surfaces Xt of genus g> 1, parameterized by the Schottky space Sg, we define a natural basis of H 0 (Xt, ω n Xt) which varies holomorphically with t and generalizes the basis of normalized abelian differentials of the first kind for n = 1. We introduce a holomorphic function F(n) on Sg which generalizes the classical product ∏∞ m=1 (1 − qm) 2 for n = 1 and g = 1. We prove the holomorphic factorization formula det ′ ∆n det N

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Year: 2004
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