Degenerations of triple coverings and Thomae's formula

2010/01/27 by Keiji Matsumoto, Matsumoto, Keiji, Tomohide Terasoma +1
Mathematics · #14H42 #32G20 #Advanced Algebra and Geometry #Advanced Combinatorial Mathematics #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #FOS: Mathematics #math.AG #msc:14H42 #msc:32G20

paper · pdf · doi:10.48550/arxiv.1001.4950

10 pages, 10 figures

openalex publication_date 2010/01/27 · arxiv created 2010/02/02 · arxiv updated 2010/02/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we prove Thomae's formula for a triple covering of \bold P¹ 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 \bold P¹. 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].

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