Abstract
A \({\mathbb{Z}_N}\) -curve is one of the form \({y^{N}=(x-\lambda_{1})^{m_{1}}\cdots(x-\lambda_{s})^{m_{s}}}\) . When N = 2 these curves are called hyperelliptic and for them Thomae proved his classical formulae linking the theta functions corresponding to their period matrices to the branching values λ1, . . . , λ s . In his work on Fermionic fields on \({\mathbb{Z}_N}\) -curves with arbitrary N, Bershadsky and Radul discovered the existence of generalized Thomae’s formulae for these curves which they wrote down explicitly in the case in which all rotation numbers m i equal 1. This work was continued by several authors and new Thomae’s type formulae for \({\mathbb{Z}_N}\) -curves with other rotation numbers m i were found. In this article we prove that for some choices of the rotation numbers the corresponding \({\mathbb{Z}_N}\) -curves do not admit such generalized Thomae’s formulae.
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González-Diez, G., Torres-Teigell, D. \({\mathbb{Z}_N}\) -Curves Possessing No Thomae Formulae of Bershadsky–Radul Type. Lett Math Phys 98, 193–205 (2011). https://doi.org/10.1007/s11005-011-0497-6
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DOI: https://doi.org/10.1007/s11005-011-0497-6