Journal of the Ramanujan Mathematical Society

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Integrality Properties of Class Polynomials for Non-Holomorphic Modular Functions


Affiliations
1 Department of Mathematics and Computer Science, Emory University, 400 Dowman Dr., W401, Atlanta, GA-30322, United States
2 Mathematics Institute, University of Cologne, Gyrhofstr, 8b 50939 Cologne, Germany
     

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In his paper Traces of Singular Moduli [14], Zagier studied values of certain modular functions at imaginary quadratic points known as singular moduli. He proved that “traces” of these algebraic integers are Fourier coefficients of certain half-integral weight modular forms. In this paper, he obtained similar results for certain non-holomorphic modular functions. However, he observed that these “singular moduli” are not necessarily algebraic integers. Based on numerical examples, the “class polynomials” whose ischolar_mains are these singular moduli seem to have predictable denominators. Here we explain this phenomenon and provide a sharp bound on these denominators.
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  • Integrality Properties of Class Polynomials for Non-Holomorphic Modular Functions

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Authors

Michael Griffin
Department of Mathematics and Computer Science, Emory University, 400 Dowman Dr., W401, Atlanta, GA-30322, United States
Larry Rolen
Mathematics Institute, University of Cologne, Gyrhofstr, 8b 50939 Cologne, Germany

Abstract


In his paper Traces of Singular Moduli [14], Zagier studied values of certain modular functions at imaginary quadratic points known as singular moduli. He proved that “traces” of these algebraic integers are Fourier coefficients of certain half-integral weight modular forms. In this paper, he obtained similar results for certain non-holomorphic modular functions. However, he observed that these “singular moduli” are not necessarily algebraic integers. Based on numerical examples, the “class polynomials” whose ischolar_mains are these singular moduli seem to have predictable denominators. Here we explain this phenomenon and provide a sharp bound on these denominators.