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Singular moduli of higher level and special cycles

  • Stephan Ehlen
Open Access
Research
  • 397 Downloads

Abstract

We describe the complex multiplication (CM) values of modular functions for \(\Gamma _0(N)\) whose divisor is given by a linear combination of Heegner divisors in terms of special cycles on the stack of CM elliptic curves. In particular, our results apply to Borcherds products of weight \(0\) for \(\Gamma _0(N)\). By working out explicit formulas for the special cycles, we obtain the prime ideal factorizations of such CM values.

Keywords

Complex multiplication CM values Special cycles Borcherds products 

Mathematics Subject Classification

Primary 11F03 Secondary 11G15 

Notes

Acknowledgments

This article contains some results that were contained in my thesis [7]. First and foremost I would like to thank my advisor Jan Hendrik Bruinier for his constant support and encouragement. I also thank Claudia Alfes and Tonghai Yang for their help. I also thank Benjamin Howard for helpful discussions and in particular for helping me with the proof of Proposition 2.5. Moreover, the helpful comments of the anonymous referee are greatly appreciated.

Role of funding source

This work is partially supported by DFG grant BR-2163/4-1.

Compliance with ethical guidelines

Competing interests The author declares that he has no competing interests.

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Copyright information

© Ehlen. 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsMcGill UniversityMontrealCanada

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