An algorithm for numerically inverting the modular j-function

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Abstract

The modular j-function is a bijective map from \(X_0(1) \setminus \{\infty \}\) to \(\mathbb {C}\). A natural question is to describe the inverse map. Gauss offered a solution to the inverse problem in terms of the arithmetic–geometric mean. This method relies on an elliptic curve model and the Gaussian hypergeometric series. Here we use the theory of polar harmonic Maass forms to solve the inverse problem by directly examining the Fourier expansion of the weight 2 polar harmonic Maass form obtained by specializing the logarithmic derivative of the denominator formula for the Monster Lie algebra.

Notes

Acknowlegements

This research was carried out in fulfillment of the requirements for the M.S. degree in mathematics at Emory University. The author would like to thank Ken Ono for his mentorship and support.

Competing interests

The authors declare that they have no competing interests.

References

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    Alwaise, E.: An algorithm for numerically computing preimages of the \(j\)-invariant. Master’s thesis, Emory University, (2017)Google Scholar
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    Bruinier, J., Kohnen, W., Ono, K.: The arithmetic of the values of modular functions and the divisors of modular forms. Compos. Math. 140(03), 552–566 (2004)MathSciNetCrossRefMATHGoogle Scholar
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    Bringmann, K., Kane, B., Löbrich, S., Ono, K., Rolen, L.: On divisors of modular forms. Preprint, arXiv:1609.08100v2 (2016)

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© SpringerNature 2018

Authors and Affiliations

  1. 1.Department of MathematicsUCLALos AngelesUSA

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