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Weber Modular Function

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Moderator: Shobhit


Post Tue Dec 31, 2013 10:31 am
Shobhit Site Admin
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Location: Jaipur, India

Define \(\displaystyle \mathfrak{f}(\tau) := \prod_{n \geq 1}\left( 1+e^{(2n-1)i\pi \tau}\right)\)

Show that \(\displaystyle \mathfrak{f}(\sqrt{-19})=\theta\) where \(\theta\) is the unique real root of \[x^3-2x-2=0\]

Explicitly,

\(\displaystyle \prod_{n \geq 1}\left( 1+e^{-(2n-1)\pi \sqrt{19}}\right)=\frac{1}{3} \left(27-3 \sqrt{57}\right)^{1/3}+\frac{\left(9+\sqrt{57}\right)^{1/3}}{3^{2/3}}\)

Post Wed Jan 01, 2014 6:42 pm

Posts: 38
Location: India, West Bengal
I have no real intuition on this, but perhaps taking log of both sides and expanding the log, interchanging the double sum and finding analogies with the thetas? You are the expert here anyways, so perhaps you can post the solution?

Post Thu Jan 02, 2014 6:55 am
Shobhit Site Admin
Site Admin

Posts: 850
Location: Jaipur, India

Actually, I haven't worked out this problem yet. :) But I feel that Kronecker's Limit formula can be exploited to prove it.

There are some similar worked examples here: http://matwbn.icm.edu.pl/ksiazki/aa/aa82/aa8245.pdf

You are more experienced with Algebraic Number Theory than me, so I think you will be able to understand it better.

Post Thu Jan 02, 2014 8:36 am

Posts: 38
Location: India, West Bengal
I am not really onto algebraic number theory, but I think I can understand those class number theoretical beasts. I will have to look at it later, thanks for the link.


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