"Half-integral weight modular forms"의 두 판 사이의 차이
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| − | <math>\Gamma_0(4)</math> | + | <math>\Gamma_0(4)</math> |
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| + | generated by <math>-I, T, ST^{-4}S</math> | ||
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| + | Define | ||
<math>\epsilon_d = \begin{cases} 1 \mbox{ if }d\equiv 1 \pmod{4} \\i \mbox{ if } d\equiv 3 \pmod{4} \end{cases}</math> | <math>\epsilon_d = \begin{cases} 1 \mbox{ if }d\equiv 1 \pmod{4} \\i \mbox{ if } d\equiv 3 \pmod{4} \end{cases}</math> | ||
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| − | <math>j(\gamma, z)= | + | Check |
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| + | <math>j(\alpha\beta,z)=j(\alpha,\beta z)j(\beta,z)</math> | ||
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| + | <math>j(\gamma, z)^2=\begin{cases} {cz+d} \mbox{ if }d\equiv 1 \pmod{4} \\ -(cz+d) \mbox{ if } d\equiv 3 \pmod{4} \end{cases}</math> | ||
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| + | <h5> </h5> | ||
2009년 8월 18일 (화) 06:02 판
\(\Gamma_0(N) = \left\{ \begin{pmatrix} a & b \\ c & d \end{pmatrix} \in SL_2(\mathbf{Z}) : \begin{pmatrix} a & b \\ c & d \end{pmatrix} \equiv \begin{pmatrix} {*} & {*} \\ 0 & {*} \end{pmatrix} \pmod{N} \right\}\)
\(\Gamma_0(4)\)
generated by \(-I, T, ST^{-4}S\)
Define
\(\epsilon_d = \begin{cases} 1 \mbox{ if }d\equiv 1 \pmod{4} \\i \mbox{ if } d\equiv 3 \pmod{4} \end{cases}\)
\(\sqrt z\) has branch in \((-\pi/2, \pi/2]\)
Define
\(j(\gamma, z)=(\frac{c}{d})\epsilon_d^{-1}\sqrt{cz+d}\) for \(\gamma \in \Gamma_0(4)\)
Check
\(j(\alpha\beta,z)=j(\alpha,\beta z)j(\beta,z)\)
\(j(\gamma, z)^2=\begin{cases} {cz+d} \mbox{ if }d\equiv 1 \pmod{4} \\ -(cz+d) \mbox{ if } d\equiv 3 \pmod{4} \end{cases}\)
W. Kohnen, Fourier coefficients of modular forms of half-integral weight. Math. Ann. 271 (1985),
237–268.
Modular functions of one variable VI
Fourier coefficients of modular forms of half-integral weight
Inventiones Mathematicae
Volume 87, Number 2 / 1987년 6월
Henryk Iwaniec