Arithmetic Properties of Traces of Singular Moduli on Congruence Subgroups

Details Arithmetic Properties of Traces of Singular Moduli on Congruence Subgroups Arithmetic Properties of Traces of Singular Moduli on Congruence Subgroups

2009-04-24

arxiv.org/abs/0904.3777v1

Mathematics

Number Theory

14 pages

Scientific paper

After Zagier proved that the traces of singular moduli $j(z)$ are Fourier coefficients of a weakly holomorphic modular form, various properties of the traces of the singular values of modular functions mostly on the full modular group $PSL_2(\mathbb{Z})$ have been investigated such as their exact formulas, limiting distribution, duality, and congruences. The purpose of this paper is to generalize these arithmetic properties of traces of singular values of a weakly holomorphic modular function on the full modular group to those on a congruence subgroup $\Gamma_0(N)$.

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