Modular Functions

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/