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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Thomae formula for abelian covers of $\mathbb {CP}^{1}$
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by Yaacov Kopeliovich and Shaul Zemel PDF
Trans. Amer. Math. Soc. 372 (2019), 7025-7069 Request permission

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
  • MR Author ID: 341444
  • Email: yaacov.kopeliovich@uconn.edu
  • Shaul Zemel
  • Affiliation: Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmund Safra Campus, Jerusalem 91904, Israel
  • MR Author ID: 914984
  • Email: zemels@math.huji.ac.il
  • 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
  • © Copyright 2019 American Mathematical Society
  • 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
  • MathSciNet review: 4024546