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### Modular Functions

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Date: 06/23/97 at 06:22:05
From: Mart de Graaf
Subject: Modular functions

Hi,

I hope you can help me with this problem. Recently I saw a documentary
on the proof of Fermat's Last Theorem. There I encountered "Modular
functions."

They said there were functions over the complex area with an
incredible amount of symmetry.

Can you tell me some more about modular functions? I got really
curious then, but I couldn't find any answers.

Thanks,
Mart
```

```
Date: 06/23/97 at 09:00:46
From: Doctor Anthony
Subject: Re: Modular functions

Dear Mart,

Modular funcions are functions with super-symmetry, which means they
can be transfomed in an infinity of different ways and yet remain
unaltered. They cannot be represented graphically because they exist
in hyperbolic space - they are complex, but with a real and imaginary
component along the x-axis and a real and imaginary component along
the y-axis.

A simple example of the type of transformation involved is:

If q = e(pi*i*w)  q is a function of w, and if now w is replaced
by w' where
aw + b
w' =  --------  ..(1)  a, b, c, d integers such that ad-bc = 1
cw + d

The infinity of transformations (1) forms a group G and then with the
aid of functions F(0), F(1), F(2), F'(1) being themselves functions of
q we can construct further functions that remain unaltered for G or
for some subgroup of G.

In the proof of Fermat's Last Theorem, use is made of the conjecture
that every elliptic equation is related to a modular form.  If an
elliptic equation is found that cannot be related to a modular form,
then that equation has no solutions.

Elliptic equations are of the form y^2 = x^3 + ax^2 + bx + c  with a,
b, c integers, and we require integer solutions for x and y.  It was
shown that if A^n + B^n = C^n with A, B, C integers was a solution of
the Fermat equation, then this could be transformed into an elliptic
equation:

y^2 = x^3 + (A^n-B^n)x^2 - A^n*B^n

It turns out that this equation can never be modular, so since all
elliptic equations are modular, we cannot have A^n + B^n = C^n

What Andrew Wiles accomplished was to prove the Taniyama-Shimura
conjecture that every elliptic equation must be modular. From there
the rest of Fermat's Last Theorem falls into place.

-Doctor Anthony,  The Math Forum
Check out our web site!  http://mathforum.org/dr.math/
```
Associated Topics:
College Imaginary/Complex Numbers

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