Degenerations of triple coverings and Thomae's formula

Abstract

In this paper, we prove Thomae's formula for a triple covering of ${\mathbf{P}}^1$ with arbitrary index. This formula gives a relation between theta constants, determinants of period integrals and the difference products of branch points. To specify a symplectic basis of the curve, we use the combinatorics of binary trees on ${\mathbf{P}}^1$. This symplectic basis behaves so well for degenerations that we obtain the absolute constant in this formula and reduce it to a special case treated in [Bershadsky-Radul], [Nakayashiki].

10 pages, 10 figures

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