Abstract
Recently, Bruinier and Ono found an algebraic formula for the partition function in terms of traces of singular moduli of a certain non-holomorphic modular function. In this paper we prove that the rational polynomial having these singular moduli as zeros is (essentially) irreducible, settling a question of Bruinier and Ono. The proof uses careful analytic estimates together with some related work of Dewar and Murty, as well as extensive numerical calculations of Sutherland.
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Acknowledgements
The authors are grateful to Kathrin Bringmann and Andrew Sutherland for many useful conversations and comments which greatly improved the exposition of this paper. They would also like to thank the anonymous referee for helpful comments. The research of the first author leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant agreement n. 335220 - AQSER. The second author thanks the University of Cologne and the DFG for his generous support via the University of Cologne postdoc grant DFG Grant D-72133-G-403-151001011.
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Mertens, M.H., Rolen, L. On class invariants for non-holomorphic modular functions and a question of Bruinier and Ono. Res. number theory 1, 4 (2015). https://doi.org/10.1007/s40993-015-0007-5
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DOI: https://doi.org/10.1007/s40993-015-0007-5