Modular Forms and Elliptic Curves: Taniyama-Shimura

Date: 10/30/97 at 01:26:38
From: Daniel Grech
Subject: Modular Forms and Elliptic Curves

Hi Dr. Math,

I watched a PBS show on Fermat's last theorem, and they kept talking 
about modular forms and elliptic curves and how they are related.  
Could you tell me what each of them are and how they relate to one 
another?

Thanks,
Dan

Date: 10/30/97 at 08:39:29
From: Doctor Jerry
Subject: Re: Modular Forms and Elliptic Curves

Hi Daniel,

Modular forms and elliptic curves are very complex.  You may wish to 
look at the comments in Eric's Treasure Trove of Mathematics at

  http://www.astro.virginia.edu/~eww6n/math/ModularForm.html   

You may be able to look up elliptic curves as well by looking around 
at this site.

-Doctor Jerry,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/   

Date: 10/30/97 at 12:15:11
From: Doctor Anthony
Subject: Re: Modular Forms and Elliptic Curves

Elliptic curves are of the form   y^2 = x^3 + ax^2 + bx + c where 
a, b, c are integers.

The problem with elliptic curves is to find if they have integer 
solutions, and if so, how many. For example the equation 
y^2 = x^3 - 2 with a = b = 0 and c = -2 has only one set of integer 
solutions, namely  x = 3, y = 5, but proving that there are no other 
solutions is extremely difficult.

The problem is simplified by making the possible numbers finite, 
i.e. working in 'clock' arithmetic. So 5-clock arithmetic uses 
only 0, 1, 2, 3, 4  the 5 = 0 again. You may recognize this as 
5 congruent 0 mod(5).  It is then possible to make progress with 
determining the number of integer solutions of the elliptic curves.

For a particular elliptic curve, the number of integer solutions in 
each clock arithmetic forms an L-series for that curve.

Example: Elliptic curve   x^3 - x^2 = y^2 + y

L-series   L1 = 1    number of solutions in 1-clock arithmetic
           L2 = 4                  "        2-clock     " 
           L3 = 4                  "        3-clock     " 
           L4 = 8                  "        4-clock     "
           L5 = 4
           L6 = 16
           L7 = 9
           L8 = 16
           .......
           .......

 
This series can go on as far as you like.  Because we cannot say how 
many solutions there are in normal number space, extending to infinity 
as it does, the L-series gives a great deal of information about the 
elliptic curve it describes. The idea is that studying the L-series 
you can learn all you want to know about its elliptic curve.

A modular form is defined by two axes, x and y, but EACH axis has a 
real and imaginary part.  In effect it is four dimensional (xr, xi, 
yr, yi) where xr means real part of x, xi means imaginary part of x, 
and similarly with yr and yi.  The four-dimensional space is called 
hyperbolic space.

The interesting thing about modular forms is that they exhibit 
infinite symmetry under transformations of the type 

               az+b
    f(z) -> f[------]
               cz+d

These are functions that remain unchanged when the complex variable z 
is changed according to the above transformation. Here the elements 
a, b, c, d, arranged as a matrix, form an algebraic group. There are 
infinitely many possible variations. They all commute with each other 
and the function f is invariant under the group of transformations.

Modular forms come in various shapes and sizes, but each one is built 
from the same basic ingredients. What differentiates each modular form 
is the amount of each ingredient it contains. The ingredients of a 
modular form are labelled from one to infinity (M1,M2,M3,....) and a 
particular modular form might contain one lot of ingredient one 
(M1=1), three lots of ingredient two (M2=3), two lots of ingredient 
three (M3=2) and so on.  So now we get an M-series

  M1 = 1
  M2 = 3
  M3 = 2
  ......
  ......       and so on.

At this point you come to the work of Taniyama and Shimura who found a 
strange affinity between some elliptic curves and some modular forms.  
However far you took the L-series and the M-series for a particular 
elliptic curve and a particular modular form, the two matched exactly.  
This led to the Taniyama-Shimura conjecture that ALL elliptic curves 
are modular.

It was in proving this conjecture that Andrew Wiles established the 
proof of Fermat's Last theorem.

The reason they are connected is as follows.

Gerhard Frey showed that IF there was a solution in integers to
x^n + y^n = z^n, say A^n + B^n = C^n then we could get an elliptic 
curve of the form
 
  y^2 = x^3 + (A^n-B^n)x^2 - (A^n.B^n)x

Another mathematician, Ken Ribet, showed that this equation could not 
be modular. So now we have the following chain of reasoning:

(1) If the Taniyama-Shimura conjecture can be proved, then every 
    elliptic curve is modular.

(2) If every elliptic curve must be modular, then the Frey elliptic 
    curve is forbidden to exist.

(3) If the the Frey elliptic curve does not exist, then there can be 
    no solutions to the Fermat equation.

(4) Therefore Fermat's Last Theorem is true.

The greatest difficulty was in proving that the Taniyama-Shimura 
conjecture was true.  This is the contribution made by Andrew Wiles, 
and the final stage in establishing Fermat's Last theorem.

-Doctor Anthony,  The Math Forum
 Check out our web site!  http://mathforum.org/dr.math/