## 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.## References

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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