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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Thomae formula for abelian covers of $ \mathbb{CP}^{1}$


Authors: Yaacov Kopeliovich and Shaul Zemel
Journal: Trans. Amer. Math. Soc. 372 (2019), 7025-7069
MSC (2010): Primary 14H42; Secondary 14H15, 32G15, 32G20, 14H81
DOI: https://doi.org/10.1090/tran/7764
Published electronically: February 11, 2019
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Abstract: Abelian covers of $ \mathbb{CP}^{1}$, with fixed Galois group $ A$, are classified, as a first step, by a discrete set of parameters. Any such cover $ X$, of genus $ g\geq 1$, say, carries a finite set of $ A$-invariant divisors of degree $ g-1$ on $ X$ that produce nonzero theta constants on $ X$. We show how to define a quotient involving a power of the theta constant on $ X$ that is associated with such a divisor $ \Delta $, some polynomial in the branching values, and a fixed determinant on $ X$ that does not depend on $ \Delta $, such that the quotient is constant on the moduli space of $ A$-covers with the given discrete parameters. This generalizes the classical formula of Thomae, as well as all of its known extensions by various authors.


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Additional Information

Yaacov Kopeliovich
Affiliation: Finance Department, School of Business, 2100 Hillside, University of Connecticut, Storrs, Connecticut 06268
Email: yaacov.kopeliovich@uconn.edu

Shaul Zemel
Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmund Safra Campus, Jerusalem 91904, Israel
Email: zemels@math.huji.ac.il

DOI: https://doi.org/10.1090/tran/7764
Received by editor(s): September 25, 2017
Received by editor(s) in revised form: July 3, 2018, and November 21, 2018
Published electronically: February 11, 2019
Article copyright: © Copyright 2019 American Mathematical Society