Journal of the
Korean Mathematical Society
JKMS

ISSN(Print) 0304-9914 ISSN(Online) 2234-3008

Article

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J. Korean Math. Soc. 1999; 36(4): 707-723

Printed July 1, 1999

Copyright © The Korean Mathematical Society.

Arithmetic of the modular function $j_4$

Chang Heon Kim and Ja Kyung Koo

Abstract

Since the modular curve $X(4)=\Gamma(4)\backslash\frak H^*$ has genus $0$, we have a field isomorphism $K(X(4))\approx\Bbb C(j_4)$ where $j_4(z)=\theta_3(\frac z2)/\theta_4(\frac z2)$ is a quotient of Jacobi theta series (\cite{Kim-Koo3}). We derive recursion formulas for the Fourier coefficients of $j_4$ and $N(j_4)$ (=the normalized generator), respectively. And we apply these modular functions to Thompson series and the construction of class fields.

Keywords: modular functions, theta series, half integral modular forms, Thompson series, class fields

MSC numbers: 11F11, 11F22, 11R04, 11R37, 14H55