An Algorithm for Numerically Computing Preimages of the j-invariant Open Access

Alwaise, Ethan James (2017)

Permanent URL: https://etd.library.emory.edu/concern/etds/0c483k319?locale=en
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Abstract

Here we explore the problem of numerically computing preimages of the j-invariant. We present an algorithm based on studying the asymptotics of the Fourier coefficients of the logarithmic derivative of j(τ). We use recent work of Bringmann, Kane, Löbrich, Ono, and Rolen, which gives asymptotics for the Fourier coefficients of divisor modular forms, to identify the real and imaginary parts of the preimage.

Table of Contents

1. Introduction and Statement of Results

2. Preliminaries on Elliptic Curves over C

3. Determining Fundamental Periods from an Elliptic Curve over C

4. Modular Forms and Harmonic Maass Forms

5. Proof of Theorem 1.0.4

6. Examples

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